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HYDEAULIC  TUEBINES 


WITH  A  CHAPTER  ON 

CENTRIFUGAL  PUMPS 


BY 

R.  L.  DAUGHERTY,  A.  B.,  M.  E. 

PROFESSOR   OF   MECHANICAL  AND   HYDRAULIC   ENGINEERING,    CALIFORNIA 

INSTITUTE  OF  TECHNOLOGY;   FORMERLY  PROFESSOR   OF   HYDRAULIC 

ENGINEERING,  RENSSELAER  POLYTECHNIC  INSTITUTE. 


THIKD  EDITION 
REVISED,  ENLARGED  AND  RESET 


IMPRESSION 


McGRAW-HILL  BOOK  COMPANY,  INC. 

NEW  YORK:    239   WEST  39TH  STREET 

LONDON:    6  &  8  BOUVERIE  ST.,  E.  C.  4 

1920 


COPYRIGHT,  1913,  1920,  BY  THE 
McGRAw-HiLL  BOOK  COMPANY,  INC. 


THE    MAPLE    PRE  S  B    YORK   PA 


PREFACE  TO  THE  THIRD  EDITION 

Since  this  book  was  first  written,  practice  has  changed  to  such 
an  extent  that  many  statements,  which  were  true  at  that  time, 
are  not  true  today.  These  portions  have  been  entirely  rewritten 
so  as  to  present  the  very  latest  features  in  construction  and 
practice.  Also  practically  every  other  chapter  has  been  altered 
and  new  matter  and  illustrations  inserted,  where  it  was  thought 
that  greater  clearness  could  be  so  attained. 

The  presentation  of  the  theory  has  been  quite  carefully 
considered  and  has  been  largely  rewritten  in  order  to  be  more 
effective.  An  attempt  has  been  made  to  so  arrange  this  that 
the  fundamental  principles  could  be  grasped  without  going  into 
a  lot  of  technical  details.  If  desired,  certain  portions  of  the 
latter,  of  theoretic  interest  only,  can  be  omitted  without  brea  ing 
the  continuity  of  thought. 

Chapters  have  been  added  on  turbine  governors  and  on  the 
methods  of  turbine  design.  The  latter  has  been  inserted  in 
order  to  meet  a  demand  for  something  on  that  subject.  The 
methods  that  are  given  are  those  employed  by  the  best  designers 
at  the  present  time.  The  procedure  avoids  the  old  "  cut  and 
try"  practice  on  .the  one  hand,  as  well  as  a  highly  theoretical 
treatment,  that  is  of  no  practical  value,  on  the  other.  It  is 
rather  a  happy  compromise  between  the  two.  The  author  is 
still  of  the  opinion  that  the  greater  number  of  engineers  are 
concerned  with  the  construction  and  operating  characteristics 
of  turbines  rather  than  with  the  details  of  their  design.  But 
there  are  some  phases  of  turbine  performance  and  construction 
that  can  be  understood  more  completely,  if  approached  from  the 
view  point  of  hydraulic  design. 

Questions  and  numerical  problems  have  been  added  at  the 
end  of  every  chapter,  in  order  to  increase  the  usefulness  of  the 
book  for  instruction  purposes.  The  questions  are  intended  to 
call  attention  to  the  most  important  features  presented  in  the 
chapter  and  also  to  bring  out  more  clearly  the  thought  there 
expressed.  The  problems  are  arranged  so  as  to  afford  applica- 
tions of  the  principles  stated  and  are  hence  quite  limited  in 
character.  If  time  available  for  the  course  permits,  it  is  thought 


415328 


vi  PREFACE 

that  problems  of  a  more  general  character  are  desirable.  But 
it  is  believed  that  it  is  better  for  the  instructor  to  prepare  these 
to  suit  his  individual  course,  and  to  vary  them  from  year  to 
year,  rather  than  to  incorporate  such  in  the  book. 

The  notation  has  been  changed  slightly  in  the  present  edition 
in  order  to  conform  more  closely  to  the  standard  notation 
recommended  by  the  Society  for  the  Promotion  of  Engineering 
Education. 

The  author  is  indebted  to  many  teachers  and  students,  who 
have  used  the  former  editions,  and  also  to  engineers  with  whom 
he  has  discussed  these  matters  for  numerous  suggestions  which 
have  been  helpful  to  him  in  the  preparation  of  the  present  volume. 

R.  L,  D. 
PASADENA,  CALIF., 
February,  1920. 


PREFACE  TO  THE  SECOND  EDITION 

In  addition  to  correcting  typographical  errors  and  rewriting 
two  articles,  the  issuing  of  a  second  edition  has  afforded  an 
opportunity  to  add  new  material  which  it  is  believed  will  increase 
the  sphere  of  usefulness  of  the  book.  The  discussion  of  several 
matters  in  the  text  has  been  amplified  and  there  have  been  added 
numerous  questions  and  problems.  This  together  with  the  15 
tables  of  test  data  in  Appendix  C  will  afford  much  suitable 
material  for  instruction  purposes. 

The  author  wishes  to  acknowledge  his  indebtedness  to  Prof. 
E.  H.  Wood  of  Cornell  University  for  his  careful  criticism  of  the 
first  edition  and  to  Prof.  W.  F.  Durand  of  Leland  Stanford 
University  for  much  valuable  assistance. 

ft.  L.  D. 

ITHACA,  N.  Y., 

August,  1914. 


vn 


PREFACE  TO  THE  FIRST  EDITION 

The  design  of  hydraulic  turbines  is  a  highly  specialized 
industry,  requiring  considerable  empirical  knowledge,  which 
can  be  aquired  only  through  experience;  but  it  is  a  subject  in 
which  comparatively  few  men  are  interested,  as  a  relatively 
small  number  are  called  upon  to  design  turbines.  But  with  the 
increasing  use  of  water  power  many  men  will  find  it  necessary 
to  become  familiar  with  the  construction  of  turbines,  understand 
their  characteristics,  and  be  able  to  make  an  intelligent  selection 
of  a  type  and  size  of  turbine  for  any  given  set  of  conditions. 
To  this  latter  class  this  book  is  largely  directed.  However,  a 
clear  understanding  of  the  theory,  as  here  presented,  ought  to 
be  of  interest  to  many  designers,  since  it  is  desirable  that  Ameri- 
can designs  be  based  more  upon  a  mathematical  analysis,  as  in 
Europe,  and  less  upon  the  old  cut  and  try  methods. 

The  br.oad  problem  of  the  development  of  water  power  is 
treated  in  a  very  general  way  so  that  the  reader  may  understand 
the  conditions  that  bear  upon  the  choice  of  a  turbine.  Thus  the 
very  important  items  of  stream  gauging  and  rating,  rainfall  and 
runoff,  storage,  etc.,  are  treated  very  briefly,  the  detailed  study 
of  these  topics  being  left  for  other  works. 

The  purpose  of  the  text  is  to  give  the  following:  A  general 
idea  of  water-power  development  and  conditions  affecting  the 
turbine  operation,  a  knowledge  of  the  principal  features  of 
construction  of  modern  turbines,  an  outline  of  the  theory  and 
the  characteristics  of  the  principal  types,  commercial  constants, 
means  of  selection  of  type  and  size  of  turbine,  cost  of  turbines 
and  water  power  and  comparison  with  cost  of  steam  power. 
A  chapter  on  centrifugal  pumps  is  also  added.  It  is  hoped  that 
the  book  may  prove  of  value  both  to  the  student  as  a  text  and 
to  the  practicing  engineer  as  a  reference.  R.  L.  D. 

CORNELL  UNIVERSITY,  ITHACA,  N.  Y., 
August,  1913. 


viu 


CONTENTS 

PAGE 

PREFACE V  .....    .    .".••.•• v 

CHAPTER  I 

INTRODUCTION 1 

Historical — The  turbine — Advantage  of  turbine  over  water  wheel — 
Advantages  of  water  wheel  over  turbine — Essentials  of  a  water- 
power  plant — Questions. 

CHAPTER  II 

TYPES  OF  TURBINES  AND  SETTINGS 7 

Classification  of  turbines— Action  of  water — Direction  of  flow — 
Position  of  shaft — Arrangement  of  runners — The  draft  tube — 
Flumes  and  penstocks — Questions. 


CHAPTER  III 

WATER  POWER 15 

Investigation — Rating  curve — The  hydrograph — Rainfall  and 
run-off — Absence  of  satisfactory  hydrograph — Variation  of  head — 
Power  of  stream — Pondage  and  load  curve — Storage — Storage  and 
turbine  selection — Power  transmitted  through  pipe  line — Pipe  line 
and  speed  regulation — Questions  and  problems. 


CHAPTER  IV 

THE  TANGENTIAL  WATER  WHEEL 30 

Development — Buckets — General  proportions — Speed   regulation 
— Conditions  of  use — Efficiency — Questions. 


CHAPTER  V 

THE  REACTION  TURBINE 41 

Development — Advantages  of  inward  flow  turbine — General  pro- 
portions of  types  of  runners — Comparison  of  types  of  runners — 
Runners — Speed  regulation — Bearings — Cases — Draft  tube  con- 
struction— Velocities — Conditions  of  use — Efficiency — Questions. 

ix 


x  CONTENTS 

CHAPTER  VI 

TURBINE  GOVERNORS 74 

General  principles — Types  of  governors — The  compensated  gover- 
nor— Questions. 

CHAPTER  VII 

GENERAL  THEORY ...*..' 80 

Equation  of  continuity — Relation  between  absolute  and  relative 
velocities — The  general  equation  of  energy — Effective  head  on 
wheel — Power  and  efficiency — Force  exerted — Force  upon  moving 
object — Torque  exerted — Power  and  head  delivered  to  runner — 
Equation  of  energy  for  relative  motion — Impulse  turbine— Reac- 
tion turbine — Effect  of  different  speeds — Forced  vortex — Free 
vortex — Theory  of  draft  tube — Questions  and  problems. 

CHAPTER  VIII 

THEORY  OF  THE  TANGENTIAL  WATER  WHEEL 105 

Introductory — The  angle  a\— ^The  ratio  of  the  radii — Force  exerted 
— Power — The  value  of  W — The  value  of  k — Constant  input — 
Variable  speed — Best  speed — Constant  speed — Variable  input — 
Observations  on  theory — Illustrative  problem — Questions  and 
problems. 

CHAPTER  IX 

THEORY  OF  THE  REACTION  TURBINE 121 

Introductory — Simple  theory — Conditions  for  maximum  efficiency 
— Determination  of  speed  for  maximum  efficiency — Losses — Rela- 
tion between  speed  and  discharge — Torque,  power,  and  efficiency — 
Variable  speed  and  constant  gate  opening — Constant  speed  and 
variable  input — Runner  discharge  conditions — Limitations  of 
theory — Effect  of  y — Questions  and  problems. 

CHAPTER  X 

TURBINE  TESTING ; 140 

Importance — Purpose  of  test — Measurement  of  head — Measure- 
ment of  water — Measurement  of  output — Working  up  results — 
Determination  of  mechanical  losses — Questions  and  problems. 

CHAPTER  XI 

GENERAL  LAWS  AND  CONSTANTS 150 

Head — Diameter  of  runner — Commercial  constants — Diameter  and 
discharge — Diameter  and  power — Specific  speed — Determintion  of 
constants — Illustrative  case — Uses  of  constants — Numerical  illustra- 
tions— Questions  and  problems. 


CONTENTS  xi 

CHAPTER  XII 

TURBINE  CHARACTERISTICS 164 

Efficiency  as  a  function  of  speed  and  gate  opening — Specific  speed 
an  index  of  type — Illustrations  of  specific  speed — Selection  of  a 
stock  turbine — Illustrative  case — Variable  load  and  head — Char- 
acteristic curve — Use  of  characteristic  curve — Questions  and 
problems. 

r"       CHAPTER  XIII 

SELECTION  OF  TYPE  OF  TURBINE      .    .    .    .    .    .    .    .    . 177 

Possible  choice — Maximum  efficiency — Efficiency  on  part-load — 
Overgate  with  high-speed  turbine — Type  of  runner  as  a  function 
of  head — Choice  of  type  for  low  head — Choice  of  type  for  medium 
head — Choice  of  type  for  high  head — Choice  of  type  for  very  high 
head — Questions  and  problems. 

CHAPTER  XIV 

COST  OF  TURBINES  AND  WATER  POWER      192 

General  considerations — Cost  of  turbines — Capital  cost  of  water 
power — Annual  cost  of  water  power — Cost  of  power  per  horse- 
power-hour— Sale  of  power — Comparison  with  steam  power — 
Value  of  water  power — Questions  and  problems. 

CHAPTER  XV 

DESIGN  OF  THE  TANGENTIAL  WATER  WHEEL 204 

General  dimensions — Nozzle  design — Pitch  of  buckets — Design 
of  buckets — Dimensions  of  case — Questions  and  problems. 

CHAPTER  XVI 

DESIGN  OF  THE  REACTION  TURBINE 211 

Introductory — General  dimensions — Profile  of  runner — Outflow 
conditions  and  clear  opening — Layout  of  vane  on  developed  cones- 
Intermediate  profiles — Pattern  maker's  sections — The  case  and 
speed  ring — The  guide  vanes — Questions  and  problems. 

CHAPTER  XVII 

CENTRIFUGAL  PUMPS 230 

Definition — Classification — Centrifugal  action — Notation — Defini- 
tion of  head  and  efficiency — Head  imparted  to  water — Losses — 
Head  of  impending  delivery — Relation  between  head,  speed  and 
discharge — Defects  of  theory — Efficiency  of  a  given  pump — 
Efficiency  of  series  of  pumps — Specific  speed  of  centrifugal  pumps 
— Conditions  of  service — Construction — Questions  and  problems. 


xii  CONTENTS 

APPENDIX  A 249 

The  retardation  curve 

APPENDIX  B .    .    .    .    .    .   251 

Stream  lines  in  curved  channels. 

APPENDIX  C .    .    .-  .    .    .    .   256 

Test  data. 

INDEX  .   279 


NOTATION 

A  =  total  area  of  streams  in  square  feet  measured  normal  to  absolute 

velocity. 
a  =  total  area  of  streams  in  square  feet  measured  normal  to  relative 

velocity. 

B  =  height  of  turbine  runner  in  inches. 
c  =  coefficient  of  discharge  in  general. 


cc  =  coefficient  of  contraction. 
cv  =  coefficient  of  velocity. 
cr  =  coefficient  of  radial  velocity. 
cu  =  coefficient  of  tangential  velocity. 
D  =  diameter  of  turbine  runner  in  inches. 
e  =  efficiency. 
eh  =  hydraulic  efficiency. 
em  =  mechanical  efficiency. 
cv  =  volumetric  efficiency. 
F  =  force  in  pounds. 
/  =  friction  factor. 

g  =  acceleration  of  gravity  in  feet  per  second  per  second. 
H  =  total  effective  head  =  z  +  V*/2g  +  p/w. 
H'  =  any  loss  in  head  in  feet. 
h  =  head  in  feet. 

h'  =  head  lost  in  friction  in  turbine  or  pump. 
h"  =  head  converted  into  mechanical  work  or  vice  versa. 
K  =  any  factor. 
K\  =  capacity  factor. 
Kz  =  power  factor. 
k  =  any  coefficient  of  loss. 
N  =  revolutions  per  minute. 
Ne  —  speed  for  maximum  efficiency. 
Ns  =  specific  speed  =  Ne\/}$.h.p./h 
m  =  abstract  number. 
n  =  abstract  number. 
0  =  axis  of  rotation.  \  ;.• 

p  =  power. 

p  =  intensity  of  pressure  in  pounds  per  square  foot. 
Q  =  total  quantity  in  cubic  feet. 
q  =  rate  of  discharge  in  cubic  feet  per  second. 
R  =  resultant  force. 
r  =  radius  in  feet. 
T  =  torque  in  foot-pounds. 
i  =  time  in  seconds. 

xiii 


xiv  NOTATION 

u  =  linear  velocity  of  a  point  on  wheel  in  feet  per  second. 
V  =  absolute  velocity  (or  relative  to  earth)  of  water  in  feet  per  second. 
Vr  —  radial  component  of  velocity  =  V  sin  or. 
Frt  =  tangential  component  of  absolute  velocity  =  V  cos  a. 

v  =  velocity  of  water  relative  to  wheel  in  feet  per  second. 
W  =  pounds  of  water  per  second  =  wq. 
w  =  density  of  water  in  pounds  per  cubic  foot. 
x  =  rt/ri. 
y  =  Ai/a2. 

a  =  angle  between  V  and  u  (measured  between  positive  directions). 
/3  =  angle  between  v  and  u  (measured  between  positive  directions). 
<£  =  ratio  Ui/\/2gh. 

<j>e  =  value  of  <f>  for  maximum  efficiency, 
to  =  angular  velocity  =  u/r. 

The  subscript  (1)  refers  to  the  point  of  entrance  and  the  subscript  (2) 
refers  to  the  point  of  outflow  in  every  case. 


HYDRAULIC  TURBINES 


CHAPTER  I 
INTRODUCTION 

1.  Historical. — Water  power  was  utilized  many  centuries 
ago  in  China,  Egypt,  and  Assyria.  The  earliest  type  of  water 
wheel  was  a  crude  form  of  the  current  wheel,  the  vanes  of  which 
dipped  down  into  the  stream  and  were  acted  upon  by  the  impact 
of  the  current  (Fig.  1).  A  large  wheel  of  this  type  was  used 
to  pump  the  water  supply  of  London  about  1581.  Such  a  wheel 
could  utilize  but  a  small  per  cent,  of  the  available  energy  of 
the  stream.  The  current  wheel,  while  very  inefficient  and 
limited  in  its  scope,  is  well  suited  for  certain  purposes  and  is 
not  yet  obsolete.  It  is  still  in  use  in  parts  of  the  United  States, 


FIG.  1. — Current  wheel. 


FIG.  2. — Breast  wheel. 


in  China,  and  elsewhere  for  pumping  small  quantities  of  water 
for  irrigation. 

The  undershot  water  wheel  was  produced  from  the  current 
wheel  by  confining  the  channel  so  that  the  water  could  not  escape 
under  or  around  the  ends  of  the  vanes.  This  form  of  wheel  was 
capable  of  an  efficiency  of  30  per  cent,  and  was  in  wide  use  up  to 
about  1800. 

The  breast  wheel  (Fig.  2)  utilized  the  weight  of  the  water 
rather  than  its  velocity  with  an  efficiency  as  high  as  65  per  cent. 
It  was  used  up  to  about  1850. 

1 


2  HYDRAULIC  TURBINES 

The  overshot  water  wheel  (Fig.  3)  also  utilized  the  weight 
of  the  water.  When  properly  constructed  it  is  capable  of 
an  efficiency  of  between  70  and  90  per  cent,  which  is  as  good 
as  the  modern  turbine.  The  overshot  water  wheel  was  exten- 
sively used  up  to  1850  when  it  began  to  be  replaced  by  the 
turbine,  but  it  is  still  used  as  it  is  well  fitted  for  some  conditions. 


-~ \ 


FIG.  3. — I.  X.  L.  steel  overshot  water  wheel.     (Made  by  Fitz  Water  Wheel  Co.) 

2.  The  Turbine. — The  turbine  will  be  more  completely 
described  in  a  later  chapter  but  in  brief  it  operates  as  follows: 
A  set  of  stationary  guide  vanes  direct  the  water  flowing  into 
the  rotating  wheel  and,  as  the  water  flows  through  the  runner, 
its  velocity  is  changed  both  in  direction  and  in  magnitude. 
Since  a  force  must  be  exerted  upon  the  water  to  change  its 
velocity  in  any  way,  it  follows  that  an  equal  and  opposite 
force  must  be  exerted  by  the  water  upon  the  vanes  of  the  wheel. 
A  turbine  may  be  defined  as  a  water  wheel  in  which  a  motion 
of  the  water  relative  to  its  buckets  is  essential  to  its  action. 


INTRODUCTION  3 

The  term  "water  wheel"  has  several  shades  of  meaning  in 
American  usage.  First  it  may  be  employed  in  its  most  general 
sense  to  indicate  any  rotary  prime  mover  operated  by  water. 
It  may  thus  be  applied  to  the  turbine,  since  the  latter  is  a  special 
type  of  water  wheel,  according  to  the  definition  in  the  preceding 
paragraph.  Second  it  may  be  used  to  designate  the  types  of 
machines  described  in  Art.  1  in  order  to  distinguish  them  from 


FIG.  4. — Francis  turbine  in  flume. 

the  modern  turbine.  Third  it  may  be  understood  to  indicate 
impulse  turbines  of  the  Pelton  type  as  contrasted  with  turbines 
of  the  reaction  type.  In  this  book  the  term  is  used  in  the  first 
or  second  sense  only,  the  context  making  it  clear  which  is  meant 
in  any  case. 

The  original  inward  flow  turbine  of  James  B.  Francis  (1849)  is 
shown  in  Figs.  4  and  5.     In  Fig.  6  are  shown  two  views  of  an 


4  HYDRAULIC  TURBINES 

inward  flow  runner  of  this  general  type  which  was  constructed 
about  1900.     This  style  is  now  obsolete. 

3.  Advantage   of  Turbine   over   Water   Wheel. — The   water 
wheel  has  been  supplanted  by  the  turbine  because : 

1.  The  latter  occupies  smaller  space. 

2.  A  higher  speed  may  be  obtained. 

3.  A  wider  range  of  speeds  is  possible. 


FIG.  5. — Francis  turbine. 

4.  It  can  be  used  under  a  wide  range  of  head,  whereas  the 
head  for  an  overshot  wheel  should  be  only  a  little  more  than 
the  diameter  of  the  wheel. 

5.  A  greater  capacity  may  be  obtained   without  excessive 
size.  . 

6.  It  can  work  submerged. 

7.  There  is  less  trouble  with  ice. 

8.  It  is  usually  cheaper. 

4.  Advantages  of  Water  Wheel  over  Turbine. — For   small 
plants  the  turbine  is  often  poorly  designed,   cheaply  made, 


INTRODUCTION  5 

unwisely  selected,  and  improperly  set.  It  may  thus  be  very 
inefficient  and  unsatisfactory.  In  such  cases  the  overshot 
water  wheel  may  be  better.  The  latter  has  a  very  high  efficiency 
when  the  water  supply  is  much  less  than  its  normal  value. 
It  is  adapted  for  heads  which  range  from  10  to  40  ft.  and  for 
quantities  of  water  from  2  to  30  cu.  ft.  per  second.1 

An  overshot  wheel  on  the  Isle  of  Man  is  72  ft.  in  diameter 
and  develops  150  h.p.     Another  at  Troy,  N.  Y.,  was  62  ft.  in 


FIG.  6. — Pure  radial  inward  flow  runner  of  the  original  Francis  type. 

diameter,  22  ft.  wide,  weighed  230  tons,  and  developed  550  h.p. 
The  latter  is  now  in  a  state  of  ruin. 

5.  Essentials  of  a  Water-power  Plant. — A  water-power  plant 
requires  some  or  all  of  the  following: 

1.  A  Storage  Reservoir. — This  may  hold  enough  water  to  run  the 
plant  for  several  months  or  more.  In  many  cases  it  may  be 
totally  lacking. 

lSee  "Test  of  Steel  Overshot  Water  Wheel,"  by  C.  R.  Weidner,  Eng. 
News,  Jan.  2,  1913,  Vol.  LXIX,  No.  1.  A  later  test  of  this  I.X.L.^wheel 
after  ball  bearings  were  substituted  gave  an  efficiency  of  92  per  cent. 


6  HYDRAULIC  TURBINES 

2.  A  Dam. — This  may  create  most  of  the  head  available  or  it 
may  merely  create  a  small  portion  of  it  and  be  erected  primarily 
to  provide  a  storage  reservoir  or  mill  pond  or  to  furnish  a  suit- 
able intake  for  the  water  conduit.     In  some  cases  the  dam  may 
be  no  more  than  a  diversion  wall  to  deflect  a  portion  of  the 
current  into  the  intake. 

3.  Intake  Equipment. — This  usually  consists  of  racks  or  screens 
to  keep  trash  from  being  carried  down  to  the  wheels  and  of  head 
gates  so  that  the  water  may  be  shut  off,  if  need  be. 

4.  The  Conduit. — The  water  may  be  conducted  by  means  of  an 
open  channel  called  a  canal  or  flume,  or  through  a  tunnel,  or  by 
means  of  a  closed  pipe  under  pressure,  which  is  called  a  penstock 
if  it  leads  direct  to  the  turbines. 

5.  The  Forebay. — A  small  equalizing  reservoir  is  often  placed 
at  the  end  of  the  conduit  from  the  main  intake  and  the  water  is 
then  led  from  this  to  the  turbines  through  the  penstock.     This  is 
called  the  forebay  and  is  also  referred  to  as  the  headwater.     In 
the  case  of  a  plant  without  any  storage  reservoir  the  body  of 
water  at  the  intake  is  often  termed  the  forebay. 

6.  The  Turbine. — The  turbine  with  its  case  or  pit  and  draft 
tube,  if  any,  comprise  the  setting. 

7.  The  Tail  Race. — The  body  of  water  into  which  the  turbine 
discharges  is  called  the  tail  water.     The  channel  conducting  the 
water  away  is  the  tail  race. 

6.  QUESTIONS 

1.  What  is  a  turbine?     What  is  a  water  wheel? 

2.  Under  what  circumstances  would  a  current  wheel  be  used?     Could 
a  turbine  be  used  under  the  same  conditions?     What  is  the  advantage  of 
the  undershot  wheel  over  the  current  wheel? 

3.  Under  what  circumstances  would  an  overshot  water  wheel  be  used? 
Could  a  turbine  be  used  under  the  same  conditions?     Could  an  overshot 
wheel  replace  any  turbine? 

4.  What  elements  would  be  found  in  every  water-power  plant?      What 
elements  may  be  in  some  and  lacking  in  others?     What  is  the  difference 
between  a  storage  reservoir  and  a  forebay? 


I       CHAPTER  II 
TYPES  OF  TURBINES  AND  SETTINGS 

7.  Classification    of   Turbines. — Turbines    are    classified    ac- 
cording to: 

1.  Action  of  Water 

(a)  Impulse  (or  pressureless) . 
(6)  Reaction  (or  pressure). 

2.  Direction  of  Flow 

(a)  Radial  outward 
(6)  Radial  inward 

(c)  Axial  (or  parallel) 

(d)  Mixed  (radial  inward  and  axial). 

3.  Position  of  Shaft 

(a)  Vertical. 
(6)  Horizontal. 

8.  Action  of  Water. — In  the  impulse  turbine  the  wheel  pas- 
sages are  never  completely  filled  with  water.     Throughout  the 
flow  the  water  is  under  atmospheric  pressure.     The  energy  of  the 
water  leaving  the  stationary  guides  and  entering  the  runner  is  all 
kinetic.     During  flow  through  the  wheel  the  absolute  velocity 
of  the  water  is  reduced  as  the  water  gives  up  its  kinetic  energy 
to  the  wheel.     In  Europe  a  type  of  impulse  turbine  commonly 
used  is  called  the  Girard  turbine.     In  the  United  States  prac- 
tically the  only  impulse  turbine  is  the  tangential  water  wheel 
or  impulse  wheel,  more  commonly  known  as  the  Pelton  wheel. 
(See  Fig.  7.) 

In  the  reaction  turbine  the  wheel  passages  are  completely  filled 
with  water  under  a  pressure  which  varies  throughout  the  flow. 
The  energy  of  the  water  leaving  the  stationary  guide  vanes  and 
entering  the  runner  is  partly  pressure  energy  and  partly  kinetic 
energy.1  During  flow  through  the  wheel  both  the  pressure 
and  the  absolute  velocity  of  the  water  are  reduced  as  the  water 
gives  up  its  energy  to  the  wheel. 

1  Strictly  speaking,  the  water  possesses  only  kinetic  energy  but  transmits 
pressure  energy. 

7 


8  HYDRAULIC  TURBINES 

Impulse  and  reaction  turbines  were  so  called  because  in  primi- 
tive types  the  force  on  the  former  was  due  to  the  "impulse" 
of  water  striking  it  while  the  force  on  the  latter  was  the  "  reac- 
tion" of  the  streams  leaving  it.  But  these  terms  are  not  very 
appropriate^for  the  forces  in  question,  since  in  either  case  the 
dynamic  force  is  due  to  a  change  produced  in  the  velocity  of 
the  water  and  the  distinction  is  largely  artificial.  And  in 
modern  turbines  the  so-called  " impulse"  at  entrance  and 
" reaction"  at  outflow  may  be  effective  in  either  type. 

A  far  better  classification  is  as  pressureless  and  pressure 
turbines.  Another  classification  is  as  partial  and  complete 
admission  turbines,  as  in  the  former  type  the  water  is  admitted 


FIG.  7. — Tangential  water  wheel  with  deflecting  nozzle. 

at  only  a  portion  of  the  circumference  while  in  the  latter  type  it 
js  necessarily  admitted  around  the  entire  circumference. 

9.  Direction  of  Flow. — Radial  flow  means  that  the  path  of  a 
particle  of  water  as  it  flows  through  the  runner  lies  in  a  plane 
which  is  perpendicular  to  the  axis  of  rotation.  If  the  water  enters 
at  the  inner  circumference  of  the  runner  and  discharges  at  the 
outer  circumference  we  have  an  outwarcf  flow  type  known  as 
the  Fourneyron  turbine.  (See  Fig.  78.) 

If  the  water  enters  at  the  outer  circumference  of  the  runner 
and  discharges  at  the  inner  circumference  we  have  an  inward 
flow  type  as  in  the  original  Francis  turbine  shown  in  Figs.  4 
and  5. 


TYPES  OF  TURBINES  AND  SETTINGS  9 

If  a  particle  of  water  remains  at  a  constant  distance  from  the 
axis  of  rotation  as  it  flows  through  the  runner  we  have  what  is 
known  as  axial  or  parallel  flow.  The  type  of  turbine  falling  in 
this  class  is  commonly  called  the  Jonval  turbine  and  is  used  to 
some  extent  in  Europe. 

If  the  water  enters  a  wheel  radially  inward  and  then  during 
its  flow  through  the  runner  turns  and  discharges  axially  we 
have  a  mixed  flow  turbine.  This  is  known  as  the  American  type 
of  turbine  and  is  also  called  a  Francis  turbine,  though  it  is  not 
identical  with  the  one  built  by  Francis. 

Modern  reaction  turbines  are  practically  all  inward  flow 
turbines  of  the  mixed  flow  type  and  to  this  type  our  discussion 
will  be  confined. 

10.  Position  of  Shaft. — The  distinction  as  to  position  of  shaft  is 
obvious.  The  vertical  shaft  turbines  are,  however,  further  classi- 
fied as  right-hand  or  left-hand  turbines  according  to  the  direction 
of  rotation.  If,  in  looking  down  upon  the  wheel  from  above,  the 
rotation  appears  clockwise  it  is  called  a  right-hand  turbine.  The 
reverse  of  this  is  a  left-hand  turbine. 

So  far  as  efficiency  of  the  runner  alone  is  concerned  there  is 
little  difference  between  vertical  and  horizontal  turbines.  Other 
things  being  equal,  the  hydraulic  losses  should  be  identical  in 
either  case,  but  there  might  be  some  difference  in  the  friction  of 
the  bearings.  As  the  latter  is  only  a  relatively  small  item,  a 
reasonable  variation  in  its  value  would  have  but  slight  effect 
on  the  efficiency. 

But  when  we  consider  the  runner  and  draft  tube  together,  we 
may  find  a  difference,  since  the  draft  tubes  are  not  necessarily 
equally  efficient  in  the  two  cases.  The  single-runner,  vertical- 
shaft  turbine,  as  shown  in  Fig.  9,  is  readily  seen  to  lend  itself  to  a 
more  efficient  draft  tube  construction  than  the  horizontal-shaft 
unit,  as  shown  in  Fig.  11,  with  the  necessary  sharp  quarter  turn 
near  the  runner  where  the  velocity  of  the  water  is  still  high. 
If  the  velocity  of  discharge  from  the  runner  is  low,  the  difference 
in  the  two  cases  may  be  insignificant,  but,  where  the  velocity  of 
the  water  is  relatively  high,  the  draft  tube  for  the  vertical-shaft 
wheel  may  be  decidedly  better. 

In  general  a  horizontal  shaft  is  more  desirable  from  the  stand- 
point of  the  station  operator  on  account  of  greater  accessibility 
and  less  bearing  trouble,  but  the  latter  is  of  less  significance 
in  recent  years  due  to  the  greater  perfection  that  has  been  ob- 


10 


HYDRAULIC  TURBINES 


tained  in  the  construction  of  suspension  bearings  for  such  service. 
A  vertical-shaft  turbine  occupies  much  less  floor  space,  but  often 
requires  more  excavation  and  a  higher  building. 

The  vertical  shaft  turbine  is  used  where  it  is  necessary  to  set 
the  turbine  down  by  the  water  while  the  generator  or  other 
machinery  that  it  drives  must  be  above.  Since  such  conditions 
are  usually  met  with  in  low-head  plants,  it  will  be  found  that 
ordinarily  the  vertical  setting  is  used  only  for  low  heads.  (See 
Fig.  8.) 


FIG.  8. — Pair  of  vertical  shaft  turbines  in  open  flume. 

The  horizontal  shaft  turbine  is  used  where  the  turbine  can  be 
set  above  the  tail  water  level  and  if  the  generator  or  other  machin- 
ery that  it  drives  can  be  set  at  the  same  elevation.  This  is  almost 
always  the  case  with  a  high-head  plant  and  is  also  quite  frequently 
the  case  with  a  low-head  plant.  (See  Fig.  10.) 

These  statements  are  purely  general  and  there  are  many 
exceptions. 


TYPES  OF  TURBINES  AND  SETTINGS 


11 


11.  Arrangement  of  Runners.— A  turbine  may  be  mounted 
up  as  an  independent  unit  with  its  own  bearings,  usually  two  in 
number,  and  connected  to  whatever  it  drives  by  means  of  a 


OOOOOOOOOIII 
OOOOOODOIIIII 


HltOOOOO'OOO 
IHOOOOOOOOO 


Fio.  9. — Vertical  shaft  turbine  with  spiral  case. 

coupling,  belt,  or  other  device.  But  some  horizontal-shaft 
hydro-electric  machines  are  set  up  as  three-bearing  units,  so  that 
neither  the  turbine  nor  the  generator  are  independent  of  each 


12 


HYDRAULIC  TURBINES 


other.  Recent  practice  is  to  reduce  this  to  two  bearings,  as  it 
is  more  compact  and  the  problem  of  alignment  is  simplified. 
The  generator  is  mounted  between  the  two  bearings  and  the  tur- 
bine runner,  which  is  relatively  light,  is  overhung  on  the  end  of 
the  generator  shaft.  Sometimes  there  are  two  runners  for  one 
generator  and  in  this  case  one  may  be  overhung  on  either  end. 
The  former  is  called  the  single-overhung  and  the  latter  the 
double-overhung  construction. 


FIG.  10. — Horizontal  shaft  turbine  in  case. 

The  double-overhung  type  is  found  only  with  horizontal-shaft 
units  and  naturally  requires  two  separate  cases  and  two  draft 
tubes.  On  the  other  hand  with  either  a  horizontal  or  vertical 
shaft  we  may  have  two  runners  discharge  into  a  common  draft 
chest  and  tube  as  in  Fig.  8.  If  open  flume  construction  is  not 
employed  this  likewise  requires  two  separate  cases.  We  may 
also  have  only  one  case  and  two  separate  draft  tubes  for  a  single 
runner  with  a  double  discharge,  as  in  Fig.  51,  page  58. 


TYPES  OF  TURBINES  AND  SETTINGS  13 

Multiple  runner  units  are  used  to  some  extent  but  present 
practice  favors  single  runners  of  larger  size  for  vertical  shaft 
installations,  as  in  Fig.  9.  For  horizontal  shafts  also  four  or 
more  runners  have  been  employed  but  the  preference  is  for  one 
runner  of  either  the  single  or  double  discharge  type  or  two  runners 
with  separate  draft  tubes. 

12.  The  Draft  Tube. — Occasionally  reaction  turbines  have 
beer  set  so  as  to  discharge  above  the  tail  water;  in  such  cases  the 


FIG.  11. — Horizontal  shaft  turbine  showing  draft  elbow. 

fall  from  the  point  of  discharge  to  the  water  was  lost.  To  avoid 
this  loss  turbines  have  been  submerged  below  the  tail  water  level 
as  in  Fig.  4,  page  3.  By  the  use  of  a  draft  tube  (or  suction 
tube),  as  in  Fig.  8  and  Fig.  10,  it  is  possible  to  set  the  turbine 
above  the  tail  water  without  suffering  any  loss  of  head.  This  is 
due  to  the  fact  that  the  pressure  at  the  upper  end  of  the  draft 
tube  is  less  than  the  atmospheric  pressure.  This  suction  com- 
pensates for  the  loss  of  pressure  at  the  point  of  entrance  to  the 
turbine  guides. 

As  will  be  shown  later,  when  the  theory  is  presented,  the  use  of 
a  draft  tube  that  diverges  or  flares  may  result  in  a  small  increase 
in  efficiency.  The  chief  advantage  of  the  draft  tube,  however,  is 


14  HYDRAULIC  TURBINES 

that  it  allows  the  turbine  to  be  set  above  the  tail  water  where  it  is 
more  accessible  and  yet  does  not  cause  any  sacrifice  in  head.  It 
is  this  that  permits  a  horizontal  shaft  turbine  to  be  installed  with- 
out any  loss  of  head. 

Since  the  wheel  passages  of  an  impulse  turbine  must  be  open 
to  the  air  it  is  readily  seen  that  the  use  of  a  draft  tube  in  the 
usual  sense  of  the  word  is  not  possible.  However,  as  will  be 
seen  later,  the  impulse  turbine  is  better  suited  for  comparatively 
high  heads  so  that  the  loss  from  the  wheel  to  the  tail  water  is 
a  relatively  unimportant  item. 

13.  Flumes  and  Penstocks. — If  the  turbine  be  used  under  a 
head  of  about  30  ft.  or  less  a  flume  may  conduct  the  water  to  an 
open  pit  as  in  Fig.  4  and  Fig.  8.  If  the  head  is  much  greater 
than  this  it  becomes  uneconomical  and  a  penstock  is  used  as  in 
Fig.  10.  The  turbine  must  then  be  enclosed  in  a  water-tight  case. 
Various  forms  of  cases  will  be  described  in  Chapter  V. 

For  penstocks  where  the  pressure  head  is  less  than  about  230 
ft.  (100  Ib.  per  square  inch)  wood-stave  pipe  is  frequently  used. 
It  is  cheaper  than  metal  pipe  for  similar  service. 

Cast-iron  pipe  is  used  for  heads  up  to  about  400  ft.  It  is  not 
good  in  large  diameters  nor  for  high  pressures  on  account  of 
porosity,  defects  in  casting,  and  low  tensile  strength.  Its 
advantages  are  durability  and  the  possibility  of  readily  obtaining 
odd  shapes  if  such  are  desired. 

For  high  heads,  steel  pipe,  either  riveted  or  welded,  is  used. 
It  is  cheaper  than  cast  iron  in  large  sizes  but  it  corrodes  more 
rapidly. 

14.  QUESTIONS 

1.  In  what  ways  may  turbines  be  classified?     How  many  of  these  are 
found  in  current  practice?     Explain  the  features  of  each. 

2.  What  are   the  differences  between  impulse  and  reaction  turbines? 
What  types  of  each  are  now  used?     Explain  the  various  directions  of  flow 
that  may  be  used. 

3.  What  are  the  relative  merits  of  horizontal  and  vertical  shaft  turbines? 
When  would  each  ordinarily  be  used? 

4.  What  arrangements  of  runners  may  we  have  for  vertical  shaft  units? 
For  horizontal  shaft  units?     What  is  meant  by  single-  and  double-over- 
hung construction? 

6.  What  two  functions  does  the  draft  tube  fulfill?  How  does  it  prevent 
loss  of  head? 


CHAPTER  III 
WATER  POWER 

15.  Investigation. — Before  a  water-power  plant  is  erected  a 
careful  study  should  be  made  of  the  stream  to  determine  the 
horse-power  that  may  be  safely  developed.     It  is  important  to 
know  not  only  the  average  flow  but  also  both  extremes.     The 
extreme  low-water  stage  and  its  duration  will  determine  the 
amount  of  storage  or  auxiliary  power  that  may  be  necessary. 
The  extreme  high-water  stage  will  fix  the  spillway  capacities  of 
dams,  determine  necessary  elevations  of  machines,  and  other 
facts  essential  to  the  safety  and  continuous  operation  of  the  plant. 

16.  Rating  Curve. — The  first  step  in  such  an  investigation  is 
the  establishment  of  a  rating  curve.    (See  Fig.  12.)    To  determine 


£ 


Cross  Section  of  Stream.  Discharge 

FIG.  12.— Rating  curve. 

the  discharge  of  the  stream  a  weir,  current  meter,  floats,  or  other 
means  may  be  employed  according  to  circumstances.1 

By  measuring  the  flow  of  the  stream  for  different  stages  a 
rating  curve  is  readily  drawn.  This  will  not  be  a  smooth  curve 
if  there  are  abrupt  changes  in  the  area  of  the  section.  A  given 
gage  height  may  really  represent  a  range  of  flows  depending  upon 
whether  the  river  is  rising  or  falling,  the  flow  being  greater  if  the 
stream  is  rising  and  less  if  it  is  falling.  This  is  because  the 
hydraulic  gradient  is  different  in  the  two  cases.2  If  possible, 

1  Hoyt  and  Grover,  "River  Discharge." 

Water  Supply  Papers  No.  94  and  No.  95  of  the  U.  S.  G.  S. 

2  Mead,  "Water-power  Engineering,"  p.  201. 

15 


16 


HYDRAULIC  TURBINES 


the  points  for  the  rating  curve  should  be  taken  when  the  river  is 
neither  rising  nor  falling.  The  discharges  from  the  rating  curve 
for  gage  readings  taken  under  all  conditions  will  be  more  or  less 
in  error,  but  in  the  end  such  errors  will  usually  balance  each  other 
and  be  unimportant. 

If  the  bed  of  the  stream  changes,  as  it  frequently  does  in 
sandy  or  alluvial  soil,  the  rating  curve  will  also  change  and  must 
be  determined  anew  from  time  to  time.  Sometimes  a  special 
permanent  control  station  may  be  constructed  to  avoid  this. 

17.  The  Hydrograph. — When  gage  readings  are  taken  regularly 
and  frequently  for  any  length  of  time  and  the  corresponding 


Dec.    Jan.      Feb.    Mar.     Apr.      May     June    July     Aug.    Sept.    Oct.      Nov. 

FIG.  13.— Hydrograph. 

discharges  secured  from  the  rating  curve  a  history  of  the  flow  may 
be  plotted  as' in  Fig.  13.  Such  a  curve  is  called  a  hydrograph. 
This  curve  is  extremely  useful  in  the  study  of  a  water-power 
proposition.  To  be  satisfactory  it  should  cover  a  period  of 
several  years  since  the  flow  will  vary  from  year  to  year.  Since 
it  is  very  important  to  know  the  extremes  also,  it  should  cover 
both  a  very  dry  year  and  a  very  wet  one  as  well  as  the  more 
normal  periods. 

18.  Rainfall  and  Run-off. — Rainfall  records  are  usually  avail- 
able for  many  years  back  and  are  a  valuable  aid  in  extending 
the  scope  of  the  hydrograph  taken,  provided  a  relation  between 
rainfall  and  run-off  can  be  estimated.  If  the  ground  be  frozen, 
or  the  slopes  steep  and  stony,  or  the  ground  saturated  and  the 


WATER  POWER 


17 


rain  violent  nearly  all  the  water  that  falls  upon  the  drainage 
basin  may  appear  in  the  stream  as  run-off.  On  the  other  hand, 
if  the  soil  be  dry  and  the  land  such  that  opportunity  is  given  it, 
all  the  rain  may  be  absorbed  and  none  of  it  appear  in  the  stream. 
Usually  the  conditions  are  such  that  the  relation  is  between  these 
two  extremes.  In  a  general  way  it  may  be  said  to  lie  between  the 
two  curves  shown  in  Fig.  14. l 

The  relation  between  rainfall  and  run-off  is  very  complicated 
and  only  partially  understood  at  present.  For  more  information 
consult  Water  Supply  Papers  of  the  U.  S.  G.  S.  and  other  sources. 


10  20  30  40  50 

Mean  Annual  Rainfall,  Depth  in  Inches 
FIG.  14. — Relation  of  rainfall  to  run-off. 

19.  Absence  of  Satisfactory  Hydrograph. — If  no  hydrograph 
of  the  stream  is  available  and  there  is  no  time  to  secure  one,  a 
study  of  the  stream  may  be  made  by  comparison  with  the  hydro- 
graphs  of  adjacent  streams.     It  is  well,  however,  to  take  a 
hydrograph  for  a  year,  if  possible,  in  order  to  be  able  to  check 
the  comparison. 

If  no  hydrographs  of  adjacent  streams  are  available,  it  is  neces- 
sary to  use  the  rainfall  records  and  make  a  thorough  study  of 
the  physical  conditions  of  the  water  shed.  If  the  relation  between 
rainfall  and  run-off  can  be  estimated,  then  fairly  satisfactory 
conclusions  may  be  drawn,  provided  a  hydrograph  for  one  year 
can  be  used  to  work  from.  Where  there  is  not  time  to  take  a 
year's  record  it  is  well  to  be  very  conservative  and  provide  for 
future  extension  of  power  if  it  is  later  found  to  be  warranted. 

20.  Variation  of  Head. — Since  the  discharge  of  any  stream  is 
usually  a  widely  varying  quantity,  it  follows  that  the  water 

1  F.  H.  Newell,  Proc,  Eng.  Club  of  Phila.,  Vol.  XII,  1895, 
2 


18 


HYDRAULIC  TURBINES 


level  at  any  point  must  vary.  If  the  turbine  be  of  the  reaction 
type  set  in  the  usual  way,  the  total  head  acting  upon  the  wheel 
will  be  the  fall  from  the  surface  of  the  head  water  to  the  surface 
of  the  tail  water  with  the  pipe  line  loss  deducted.  If,  in  times 
of  high  water,  the  head  water  level  rose  the  same  amount  as  the 
tail  water  level  the  net  head  under  which  the  turbine  operated 
would  remain  constant.  But,  under  the  usual  conditions,  the 
tail  water  level  rises  more  than  the  head  water  level  and  the  net 
head  under  which  the  turbine  operates  becomes  less.  This  is 
illustrated  in  Fig.  15  where  three  rates  of  flow  are^shown. 

At  high  water  the  horse-power  of  the  stream  may  be  large  even 
though  the  fall  be  reduced,  owing  to  the  increased  quantity  of 


FIG.  15. — Decrease  of  available  head  at  high  water. 

water.  But  the  horse-power  of  the  turbine  may  be  seriously 
diminished.  A  turbine  is  only  a  special  form  of  orifice  and  there- 
fore the  discharge  through  it  is  proportional  to  the  square  root  of 
the  head.  If  then  the  discharge  through  it  be  reduced  due  to  the 
lower  head,  the  horse-power  input  to  the  turbine  is  decreased.  If 
the  best  efficiency  is  to  be  obtained,  the  speed  also  should  vary  as 
the  square  root  of  the  head.  But  usually  the  turbine  is  compelled 
to  run  at  constant  speed  and  this  causes  a  further  reduction  of  the 
power  of  the  turbine  since  the  efficiency  is  lowered.  (The  speed 
should  be  the  best  for  low  water  because  economy  of  water  is  then 
important.)  It  is  thus  seen  that  the  decrease  of  the  head  at  high 
water  causes  a  loss  of  power  and  a  drop  in  efficiency.  This 
change  of  head  will  be  an  insignificant  item  for  a  high-head  plant 
but  may  be  very  serious  for  a  low-head  plant. 


WATER  POWER 


19 


21.  Power  of  Stream. — If  the  conditions  are  such  that  there 
is  no  appreciable  change  in  head,  the  hydrograph  with  a  suitable 
scale  may  represent  the  power  of  the  stream  also.  But  if  the  head 
varies  to  any  extent  with  the  flow  then  the  power  curve  must  be 
computed  from  the  hydrograph  by  using  the  heads  that  would  be 


Dec.   Jan.  leb.  Mar.  Apr.    May  June  July  Aug.  Sept.  Oct.  Nov. 
FIG.  16. — Power  curve  of  a  stream. 

obtained  at  various  stages  of  flow.  Or  the  hydrograph  itself  may 
still  be  used  as  a  power  curve  if  the  power  scale  that  is  used  is 
made  to  vary  as  the  head  varies  instead  of  being  uniform. 

If  Fig.  16  represents  the  power  curve  of  a  stream  then  A-B 
represents  the  greatest  power  that  the  stream  can  be  counted 
upon  to  furnish  at  all  times. 


Average  Load 


\ 


12 


68       10 


12 

M 

Time 

FlG.    17. 


6        8        10       12 


22.  Pondage  and  Load  Curve. — By  pondage  is  meant  the 
storing  of  a  limited  amount  of  water.  If  the  plant  be  operated 
24  hours  on  a  steady  load  then  pondage  is  of  little  value  except  for 
equalizing  the  flow  of  water  when  the  stream  is  low.  But  if  the 


20  HYDRAULIC  TURBINES 

plant  be  operated  for  only  a  portion  of  the  24  hours  or  if  the  load 
be  variable  as  shown  by  the  load  curve  in  Fig.  17,  then  the  water 
that  is  not  used  when  the  load  is  light  may  be  stored  and  used 
when  the  load  is  heavy.  If  the  pondage  be  ample,  the  average 
load  carried  by  the  plant  may  then  be  equal  to  A-B  in  Fig.  16, 
while  the  peak  load  may  be  much  greater. 

23.  Storage. — By  storage  is  meant  the  storing  of  a  consider- 
able quantity  of  water,  so  that  it  varies  from  pondage  in  degree 
only.     Pondage  indicates  merely  sufficient  capacity  to  supply 
water  for  a  few  hours  or  perhaps  a  few  days,  but  storage  implies 
a  capacity  which  can  supply  water  needed  during  a  dry  spell  of 
several  weeks  or  months  or  more.     The  effect  of  storage  is  to 
enable  the  minimum  power  of  the  stream  to  be  raised  from  A-B 
to  C-D  (Fig.  16).     The  greater  the  storage  capacity  the  higher 
C-D  is  placed  until  it  equals  the  average  power  of  the  stream. 
The  water  for  the  turbines  may  be  drawn  direct  from  the  storage 
reservoir  (in  which  case  the  head  varies)  or  the  reservoir  may  be 
used  as  a  stream  feeder  only. 

A  plant  operating  under  a  low  head  requires  a  relatively  large 
amount  of  water  for  a  given  amount  of  power.  A  storage  basin 
for  such  a  plant  would  require  a  very  large  capacity  if  it  were  to 
furnish  power  for  any  length  of  time.  But  a  low  head  is  usually 
found  in  a  fairly  flat  country  where  it  is  possible  to  construct  a 
storage  reservoir  of  limited  capacity  only,  and  often  none  at  all, 
on  account  of  flooding  the  surrounding  country.  But  for  a  high 
head  the  conditions  are  different  as  only  a  relatively  small  amount 
of  water  is  required  so  that  the  capacity  of  the  storage  reservoir 
need  not  be  excessive.  The  higher  the  head,  the  more  valuable  a 
cubic  foot  of  water  becomes.  The  topography  of  a  country  where 
a  high  head  can  be  developed  is  usually  such  that  storage  reser- 
voirs of  large  capacity  can  be  constructed  at  reasonable  cost.  A 
low-head  plant  usually  possesses  pondage  only — a  high-head 
plant  usually  possesses  storage. 

24.  Storage   and   Turbine    Selection. — If   a   plant   possesses 
neither  storage  nor  pondage,  or  the  stream  flow  may  not  be  inter- 
rupted because  of  other  water  rights,  the  economy  of  water  when 
the  turbine  is  running  under  part  load  is  of  no  importance.     The 
efficiency  at  full  load  is  all  that  is  of  interest.     But  if  the  plant 
does  have  pondage  or  storage  in  any  degree  the  economy  of  water 
under   all   loads   is   of  importance.     The   more    extensive    the 
pondage  the  more  valuable  a  high  efficiency  on  all  loads  becomes. 


WATER  POWER 


21 


Thus  the  question  of  storage  has  an  important  bearing  in  turbine 
selection. 

25.  Power  Transmitted  through  Pipe  Line. — Suppose  that 
a  nozzle,  whose  area  can  be  varied,  is  placed  at  the  end  of  a  pipe 
line  B-C  (Fig.  18).  With  the  nozzle  closed  we  have  a  pressure 
head  at  C  of  CX  which  is  equal  to  the  static  head.  The  hydraulic 


FIG.  18. — Varying  rates  of  flow  in  pipe  line. 

gradient  is  then  a  horizontal  line.  If  the  nozzle  be  partially 
opened,  so  that  flow  takes  place,  the  losses  in  the  pipe  line  as  well 
as  the  velocity  head  in  the  pipe  cause  the  pressure  to  drop  to 
CY.  A  further  opening  of  the  nozzle  would  cause  the  pressure 
to  drop  to  a  lower  value.  If  the  nozzle  were  removed  the  pres- 


FIG.  19 


««-— ^— ^—— ^^^^— « 

Rate  of  Discharge 
. — Head  and  power  at  end  of  pipe  line. 


sure  at  C  is  then  atmospheric  only,  which  we  ordinarily  call  zero 
pressure.     The  hydraulic  gradient  is  then  A-C. 

Head  is  the  amount  of  energy  per  pound  of  water.  The  head  at 
C  is  the  elevation  head,  taken  as  zero,  plus  the  pressure  head,  plus 
the  velocity  head.  When  the  discharge  is  zero  the  head  is  a 
maximum,  being  equal  to  CX.  When  the  nozzle  is  removed  the 
discharge  is  a  maximum  but  the  head  at  C  is  a  minimum,  being 


22  HYDRAULIC  TURBINES 

only  the  velocity  head.  For  any  intermediate  value  of  discharge 
the  head  will  be  intermediate  between  these  two  extremes. 

The  power  transmitted  through  the  pipe  line  and  delivered  at  C 
is  a  function  of  both  the  quantity  of  water  and  the  head.  It  is 
zero  when  the  discharge  is  zero  and  very  small  when  the  discharge 
is  a  maximum.  The  power  becomes  a  maximum  for  a  discharge 
between  these  two  extremes  as  is  shown  in  Fig.  19.  Let  the 
rate  of  discharge  through  the  pipe  be  denoted  by  q,  the  net 
head  at  C  by  h,  the  loss  of  head  by  H',  and  the  height  CX  by  z. 
If  the  loss  of  head  in  the  pipe  be  assumed  proportional  to  the 
square  of  the  velocity  of  flow  we  may  write  Hf  =  Kq2,  where  K 
is  a  constant  whose  value  depends  upon  the  length,  size,  and 
nature  of  the  pipe.  Then 

Power  =  qh  =  q(z  -  H')  =  qz  -  Kq* 
Differentiating     d(PowQv)/dq  =  z  -  ZKq2  =  0 
Or  z  =  3Kqz  =  3H'. 

Thus  the  power  delivered  by  a  given  pipe  line  is  a  maximum  when 
the  flow  of  water  is  such  that  one-third  the  head  available  is 
used  up  in  pipe  friction,  leaving  the  net  head  only  two-thirds 
of  that  available. 

The  efficiency  of  the  pipe  line  is  expressed  by  h/z.  Thus  in  the 
case  where  the  pipe  line  is  delivering  its  maximum  power,  its 
efficiency  is  only  66%  per  cent.  But  if  economy  in  the  use  of 
water  is  an  object  the  discharge  through  the  pipe  would  be  kept 
at  a  lower  value  than  this  so  as  to  prevent  so  much  of  the  energy 
of  the  water  being  wasted.  For  a  given  quantity  of  water,  this 
means  that  a  larger  pipe  would  be  used,  so  that  its  efficiency 
would  be  higher.  In  a  similar  manner,  if  a  given  amount  of  power 
is  required,  the  smallest  pipe  that  can  be  used  will  be  of  such  a 
size  that  its  efficiency  is  66%  per  cent.  As  the  pipe  is  made 
larger  than  this,  its  efficiency  rises  and  the  amount  of  water 
required  decreases.1 

The  most  economical  size  of  pipe  may  be  found  as  shown  in 
Fig.  20.  One  curve  represents  the  annual  value  of  the  power  lost 

1  It  should  be  noted  that  in  this  paragraph  there  are  three  separate  cases 
mentioned.  First  the  size  of  the  pipe  is  fixed  and  different  rates  of  discharge 
are  assumed  to  flow  through  it.  Second  the  quantity  of  water  available  is 
fixed  and  the  size  of  the  pipe  is  the  variable.  Third  the  power  delivered  is 
fixed  and  the  size  of  the  pipe  is  varied, 


WATER  POWER 


23 


in  pipe  friction,  the  other  the  annual  fixed  charge  on  the  pipe. 
This  includes  interest  on  the  money  expended,  depreciation, 
repairs,  etc.  The  total  cost  of  the  pipe  per  year  is  the  curve 
whose  ordinates  are  the  sums  of  the  other  two.  The  size  of  pipe 
for  which  this  sum  is  a  minimum  is  the  most  economical. 

If  the  rate  of  discharge  is  not  constant,  careful  study  must 
be  made  of  the  load  curve  in  order  to  determine  what  value  of 
the  rate  of  discharge  will  give  the  average  power  lost.  For  the 
typical  load  curve  this  value  may  often  be  found  to  be  about  80 
per  cent,  of  the  maximum  flow. 


Size  of  Pipe 
FIG.  20. — Determination  of  economic  size  of  pipe. 

It  must  be  noted  that  this  solution  may  not  always  be  the  most 
practical  because  of  other  considerations.  For  instance  the 
velocity  of  the  water  may  be  too  high  and  thus  give  rise  to 
trouble  due  to  water  hammer.  Again  if  the  loss  of  head  is  too 
large  a  percentage  of  the  head  available,  the  variation  of  the  net 
head  between  full  discharge  and  no  discharge  may  be  con- 
siderable. This  might  cause  trouble  in  governing  the  turbine. 

26.  Pipe  Line  and  Speed  Regulation. — A  fundamental  propo- 
sition in  mechanics  is  that 

input  =  output  -f  losses  +  gain  in  energy. 

If  the  speed  of  a  turbine  is  to  remain  constant  it  follows  that  the 
input  must  aways  be  equal  to  the  power  output  plus  the  losses. 
As  the  power  output  varies,  therefore,  the  quantity  of  water  sup- 


24  HYDRAULIC  TURBINES 

plied  to  the  turbine  must  vary.  It  is  thus  apparent  that  a  turbine 
does  not  run  under  an  absolutely  constant  head  at  all  loads. 
By  referring  to  Fig.  19  it  is  seen  that  when  the  turbine  is  using 
only  a  small  quantity  of  water  the  head  will  be  higher  than  when 
it  is  carrying  full  load. 

If  the  load  on  a  turbine  is  rapidly  reduced  the  quantity  of  water 
supplied  to  it  must  be  very  quickly  decreased  in  order  to  keep  the 
speed  variation  small.  This  means  that  the  momentum  of  the 
entire  mass  of  water  in  the  penstock  and  draft  tube  must  be  sud- 
denly diminished.  If  the  penstock  be  long  a  big  rise  in  pressure 
may  be  produced  so  that  momentarily  the  pressure  may  be  greater 
than  the  static  pressure.  This  increase  in  pressure  may  be  suffi- 
cient to  even  cause  an  increase  in  the  power  input  for  a  very  brief 
interval  of  time.  On  the  other  hand,  if  the  load  on  the  turbine  be 
suddenly  increased,  the  water  in  the  penstock  and  draft  tube  must 
be  accelerated  and  this  causes  a  temporary  drop  in  pressure  below 
the  normal  value,  and  for  the  time  being  the  power  input  to  the 
turbine  may  be  diminished  below  its  former  value.  The  longer 
the  pipe  line  and  the  higher  the  maximum  velocity  of  flow,  the 
worse  these  effects  become.  It  is  thus  seen  that  the  speed  regu- 
lation depends  upon  the  penstock  and  draft  tube  as  well  as  upon 
the  governor  and  the  turbine.1 

If  the  velocity  of  the  water  is  checked  too  suddenly  a  dangerous 
water  hammer  may  be  produced.  In  order  to  avoid  an  excessive 
rise  in  pressure,  relief  valves  are  often  provided.  Automatic  re- 
lief valves  are  analogous  to  safety  valves  on  boilers;  they  do  not 
open  until  a  certain  pressure  has  been  attained.  Mechanically 
operated  relief  valves  are  opened  by  the  governor  at  the  same  time 
the  turbine  gates  are  closed  and  afford  the  water  a  by-pass  so  that 
there  is  no  sudden  reduction  of  flow.  To  prevent  waste  of  water 
these  by-passes  may  be  slowly  closed  by  some  auxiliary  device. 
Another  means  of  equalizing  these  pressure  variations  is  to  place 
near  the  turbine  a  stand  pipe  or  a  surge  chamber,  with  compressed 
air  in  its  upper  portion,  or  open  to  the  atmosphere  if  it  can  be 
made  high  enough.  These  have  the  advantage  over  the  re- 
lief valves  that  they  are  not  only  able  to  prevent  the  pressure  in- 

1  A  case  may  be  cited  where  the  length  of  a  conduit  was  7.76  miles,  the 
average  cross-section  100  sq.  ft.,  and  the  maximum  velocity  10  ft.  per  second. 
The  amount  of  water  in  the  conduit  was,  therefore,  128,125  tons  and  with 
the  velocity  of  10  ft.  per  second  there  would  be  in  round  numbers  200,000 
ft  .-tons  of  kinetic  energy. 


WATER  POWER  25 

crease  from  being  excessive  but  they  are  able  to  supply  water  in 
case  of  an  increasing  demand  and  thus  prevent  too  big  a  pressure 
drop.1 

27.  QUESTIONS  AND  PROBLEMS 

1.  Before  a  water  power  plant  is  built  what  information  should  be  ob- 
tained regarding  the  stream?     How  may  this  be  determined? 

2.  What  is  the  rating  curve  of  a  stream?     How  is  it  obtained?     What  use 
is  made  of  it?     Is  it  always  the  same? 

3.  What  is  the  hydrograph?     How  is  it  obtained?     What  is  its  use? 

4.  What  use  may  be  made  of  rainfall  records,  if  a  hydrograph  of  the  stream 
has  been  obtained  by  direct  measurement?     What  use  may  be  made  of 
rainfall  records,  if  no  hydrograph  is  in  existence? 

5.  Is  the  head  on  a  water  power  plant  constant?     What  causes   this? 
Do  the  head  water  levels  and  the  tail  water  levels  change  at  the  same  rate? 
Why?     What  effect  does  this  have  on  the  power  and  efficiency  of  the  tur- 
bine?    What  types  of  plants  are  most  seriously  affected? 

6.  How  is  the  power  of  a  stream  to  be  determined?     What  effect  does 
pondage  have  upon  this?     What  is  the  difference  between  pondage  and 
storage  and  how  do  they  differ  in  their  effects  upon  the  extent  of  the  power 
development? 

7.  As  the  flow  of  water  through  a  given  pipe  increases,  how  do  the  head 
and  power  delivered  change?     How  does  the  efficiency  vary?     For  what 
condition  is  the  power  a  maximum?     Is  this  desirable? 

8.  If  a  given  rate  of  discharge  is  to  be  used  for  power,  how  may  the 
proper  size  of  pipe  be  determined?     Are  there  several  factors  that  need  to 
be  considered? 

9.  If  a  given  amount  of  power  is  required  and  the  water  supply  is  ample, 
how  can  the  smallest  size  of  pipe  that  would  serve  be  found?     What  would 
limit  the  largest  size  that  might  be  used? 

10.  How  does  the  head  on  a  turbine  change  with  the  load  the  wheel 
carries?     What  effect  does  the  pipe  line  have  upon  speed  regulation? 

11.  What  devices  are  employed  to  care  for  the  condition  when  the  gover- 
nor suddenly  diminishes  the  water  supply?     What  may  be  used  to  care  for 
a  sudden  demand? 

12.  The  following  table  gives  the  results  of  a  current  meter  traverse  of  a 
stream:  Velocity  of  water  in  ft.  per  second  equals  2.2  times  revolutions  per 
second  of  the  metre  plus  0.03. 

From  this  data  compute  the  area,  rate  of  discharge,  and  mean  velocity  of 
the  stream.  (The  mean  velocity  in  a  vertical  ordinate  will  be  found  at 
about  0.6  the  depth.  The  mean  velocity  is  obtained  with  a  slightly  greater 
degree  of  accuracy  by  taking  the  mean  of  readings  at  0.2  and  0.8  the  depth. 

1  See  "Control  of  Surges  in  Water  Conduits,"  by  W.  F.  Durand,  Journal 
A.  S.  M.  E.,  June,  1911;  "The  Differential  Surge  Tank,"  by  R.  D.  John- 
son, Trans.  A.  S.  C.  E.,  Vol.  78,  p.  760,  1915;  and  "Pressure  in  Penstocks 
caused  by  the  Gradual  Closure  of  Turbine  Gates,"  by  N.  R.  Gibson,  Proc. 
A.  S.  C.  E.,  Vol.  45,  Apr.,  1919. 


26 


HYDRAULIC  TURBINES 


Distance 
from 
initial 
point 

Depth 
of 
stream 

Depth 
of  obser- 
vation 

Time 
in 
seconds 

Revolu- 
tions 

Distance 
from 
initial 
point 

Depth 
of 
stream 

Depth 
of  obser- 
vation 

Time 
in 
seconds 

Revolu- 
tions 

2 

0.0 

70 

1.4 

0.28 

40 

20 

5 

0.7 

0.42 

60 

10 

1.12 

43 

10 

10 

1.0 

0.60 

48 

10 

75 

1.2 

0.24 

57 

30 

15 

1  .0 

0.60 

48 

15 

0.96 

50 

15 

20 

0.9 

0.54 

48 

20 

80 

1.3 

0.26 

51 

20 

25 

1.5 

0.30 

48 

20 

1.04 

44' 

10 

1.20 

42 

15 

85 

1.4 

0.28 

52 

20 

30 

1.7 

0.34 

41 

30 

1.12 

43 

10 

1.36 

48 

30 

90 

1.2 

0.24 

49 

20 

35 

1.9 

0.38 

45 

30 

0.96 

53 

15 

1.52 

50 

20 

95 

1.3 

0.26 

40 

15 

40 

1.8 

0.36 

45 

30 

1.04 

39 

10 

1.44 

43 

20 

100 

1.1 

0.22 

45 

20 

45 

1.7 

0.34 

49 

30 

0.88 

56 

15 

1.36 

45 

20 

105 

1.0 

0.20 

45 

20 

50 

1.6 

0.32 

42 

30 

0.80 

55 

15 

1.28 

43 

20 

110 

1.2 

0.24 

46 

20 

55 

1.5 

0.30 

50 

30 

0.96 

59 

10 

1.20 

49 

20 

115 

1.2 

0.24 

41 

15 

60 

1.6 

0.32 

53 

30 

0.96 

58 

10 

1.28 

52 

15 

120 

0.8 

0.48 

55 

5 

65 

1.4 

0.28 

55 

30 

125 

0.9 

0.54 

47 

5 

1  .12 

55 

15 

130 

1  .1 

0.66 

42 

2 

135 

1  .1 

140 

0.0 

The  area  between  two  ordinates  may  be  taken  as  the  product  of  the  distance 
between  them  by  half  the  sum  of  the  two  depths.  The  mean  velocity  in 
such  an  area  may  be  taken  as  half  the  sum  of  the  mean  velocities  of  the 
ordinates.  The  product  of  area  and  mean  velocity  gives  the  discharge 
through  the  area.  The  sum  of  all  such  partial  areas  and  discharges  gives 
the  total  area  and  total  discharge  of  the  stream.  The  total  discharge  divided 
by  the  total  area  gives  the  mean  velocity  of  the  stream.) 

Ans.     171.8  sq.  ft.,  146.8  cu.  ft.  per  second. 

13.  The  traverse  of  the  stream  given  in  problem  (12)  was  made  May  14, 
1913  when  the  gage  height  was  1.21  ft.  Other  ratings  had  been  made  as 
noted. 


Date 

Width, 
ft. 

Area, 
sq.  ft. 

Mean 
velocity, 
ft.  per  sec. 

Gage 
height, 

ft. 

Discharge, 
sec.-ft. 

November  3,  1906  
May  10,  1908 

138 
138 

485 
345 



3.10 
2  32 

1345 

758 

September  4,  1908  

60 

45 

0.57 

24 

July  24,  1909  

138 

157 

1.12 

150 

November  19,  1909  

72 

60 

0.74 

51 

May  12,  1910  

138 

226 

1.63 

364 

October  11,  1911  . 

138 

165 

1  32 

202 

July  30,  1912.  .      . 

78 

49 

0  70 

42 

May  14,  1913 

138 

172 

1  21 

WATER  POWER 


27 


It  will  be  noted  that  the  data  is  not  always  consistent,  due  to  changes  in 
the  bed  of  the  stream.  From  the  data  given  draw  to  scale  the  probable  out- 
line of  the  cross-section  of  the  stream.  Plot  values  of  area,  mean  velocity, 
and  discharge  against  gage  height.  (The  area  and  velocity  curves  can  be 
extended  with  greater  assurance  than  the  discharge  curve.  By  computing 
values  of  discharge  from  these  two,  the  discharge  curve  may  be  produced 
beyond  readings  taken.) 

14.  The  daily  gage  heights. of  the  stream  of  the  preceding  problem  for 
1912  are  given  below.  Plot  the  hydrograph.  Note  values  of  maximum, 
minimum,  and  average  flow,  and  the  duration  of  the  minimum  flow. 


Day 

Jan. 

Feb. 

Mar. 

Apr. 

May 

June 

July 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 

1 

1.10 

.48 

6.50 

2.30 

2.70 

1.32 

0.89 

0.80 

1.04 

1.22 

1.48 

1.31 

2 

1.20 

.46 

5.70 

2.90 

2.30 

1.25 

0.85 

0.74 

1.25 

1.24 

.60 

1.34 

3 

1.10 

.48 

4.60 

3.80 

2.20 

2.30 

0.82 

0.76 

1.34 

1.16 

.48 

2.30 

4 

1.30 

.38 

2.00 

2.90 

2.16 

0.90 

0.79 

1.14 

1.12 

.42 

1.90 

5 

1.05 

.35 

1.80 

2.60 

1.90 

.55 

0.86 

0.74 

1.06 

.14 

.31 

1.70 

6 

1.10 

.32 

1.55 

2.40 

1.85 

.50 

0.84 

0.71 

1.01 

.09 

.28 

2.10 

7 

2.20 

.30 

1.15 

2.35 

1.95 

.80 

0.80 

0.71 

0.92 

.01 

.31 

2.10 

8 

2.55 

.52 

1.04 

2.90 

2.15 

.55 

0.76 

0.71 

0.02 

.95 

.00 

1.60 

9 

2.30 

.58 

1.90 

2.45 

2.40 

.38 

0.74 

0.74 

0.95 

.14 

.70 

1.70 

10 

2.50 

1.58 

1.95 

2.25 

2.20 

.29 

0.72 

0.79 

0.91 

.05 

.55 

1.65 

11 

1.85 

1.44 

1.70 

2.15 

1.95 

.24 

0.74 

2.35 

0.92 

.09 

.50 

1.55 

12 

2.00 

1.48 

1.70 

2.10 

1.80 

.21 

0.74 

1.50 

0.90 

.10 

.41 

1.49 

13 

1.85 

1.52 

3.60 

2.05 

1.80 

.21 

0.72 

1.05 

0.90 

.11 

.42 

1.35 

14 

1.85 

1.60 

3.10 

2.00 

1.70 

.15 

0.88 

1.20 

0.85 

.08 

.50 

1.48 

15 

2.00 

1.50 

3.20 

2.05 

2.00 

.12 

0.88 

1.00 

0.85 

.01 

.46 

1.35 

16 

1.80 

1.52 

4.50 

2.30 

.15 

0.83 

0.95 

0.88 

0.95 

.39 

1.40 

17 

1.75 

1.55 

3.20 

2.35 

2.00 

.09 

0.80 

1.42 

0.86 

0.89 

.38 

1.38 

18 

1.85 

1.58 

3.00 

3.00 

1.80 

.11 

1.00 

1.15 

0.94 

0.91 

.94 

1.34 

19 

2.20 

1.56 

2.80 

3.06 

1.70 

.05 

0.98 

1.42 

1.15 

1.01 

.28 

2.25 

20 

2.80 

1.55 

2.60 

2.70 

1.65 

.02 

0.84 

1.30 

1.14 

0.96 

.38 

2.10 

21 

2.86 

1.58 

2.25 

2.35 

1.60 

.06 

0.85 

1.12 

1.01 

0.96 

.28 

1.90 

22 

6.30 

2.42 

2.30 

2.10 

1.60 

1.02 

1.02 

1.20 

1.04 

0.92 

1.26 

1.75 

23 

3.35 

2.29 

1.95 

2.25 

1.48 

0.94 

0.95 

1.06 

1.01 

1.35 

1.22 

1.70 

24 

2.55 

2.30 

2.05 

2.10 

1.43 

0.95 

0.76 

1.14 

1.18 

2.80 

1.36 

1.60 

25 

2.16 

2.22 

2.22 

2.10 

.45 

0.95 

0.84 

1.04 

2.05 

2.60 

1.70 

1.55 

26 

1.95 

2.20 

1.95 

1.90 

.70 

0.94 

0.84 

0.99 

1.70 

2.25 

1.60 

1.60 

27 

1.95 

2.05 

2.00 

2.00 

.55 

0.98 

0.81 

0.96 

1.45 

.95 

1.50 

1.70 

28 

1.80 

2.60 

2.20 

1.90 

.40 

0.88 

0.80 

1.04 

1.32 

.80 

1.42 

1.80 

29 

1  68 

2  60 

4.20 

1.80 

42 

0.90 

0.79 

1.00 

1.28 

.70 

1.40 

1.65 

30 

1.60 

3.60 

3.20 

.65 

0.89 

0.75 

0.91 

1.26 

.60 

1.35 

1.70 

31 

1.62 

3.00 

.38 

0.72 

0.99 

.50 

2.50 

15.  The  following  table  gives  the  rainfall  record  in  a  certain  vicinity  for 

several  years,  and  also  the  estimated  run-off.     The  relation  of  rainfall  to 

run-off  is  not  only  different  for  different  drainage  basins,  but  for  a  given 

drainage  basin  it  varies  according  to  the  time  of  year  and  the  extent  of  the 

rainfall.     There  is  thus  no  constant  relation  between  the  two  in  the  table.1 

With  these  records  construct  a  hydrograph  for  the   estimated  average 

1  See  Kuichling's  Rainfall- Run-off  Diagrams  in  the  report  on  the  New 

York  State  Barge  Canal  of  1900. 


28 


HYDRAULIC  TURBINES 


monthly  rate  of  discharge  of  a  stream  with  a  drainage  basin  of  20  square 
miles. 


19 

07 

19 

08 

19 

09 

19 

10 

19 

11 

19 

12 

Rainfall, 

inches 

Run-off, 
inches 

Rainfall, 
inches 

Run-off, 
inches 

Rainfall, 
inches 

Run-off, 
inches 

Rainfall, 
inches 

5e"« 
|| 

C  o 
3  fl 
PS"" 

Rainfall, 

inches 

Run-off, 
inches 

3"  2 

!§•§ 

c3  fi 
P»-- 

*?I 

C  o 

.2  a 
tf"1 

Jan  

3.05 

1.9 

3.21 

2.0 

4.14 

2.2 

1.15 

10 

2.85 

1.8 

4.91 

3.5 

Feb  

1.95 

1.8 

4.61 

3.5 

5.17 

4.0 

1.84 

1.7 

2.11 

1.8 

4.04 

3.0 

March.  .  .  . 

1.91 

2.6 

4.04 

4.0 

3.74 

3.6 

1.48 

2.3 

2.98 

3.2 

5.16 

4.7 

April  

2.19 

2.2 

3.78 

3.1 

4.91 

3.6 

5.96 

4.0 

2.82 

2.6 

5.71 

3.9 

May  

2.72 

1.2 

4  .98 

2.0 

2.94 

1  .3 

2.58 

1.2 

1.33 

0.9 

3.15 

1  .3 

June  

2.73 

0.8 

1.53 

0.6 

3.50 

0.9 

3.47 

0.9 

7.98 

2.3 

1.32 

0.6 

July  

2.74 

0.4 

3.44 

0.4 

1.86 

0.3 

2.00 

0.3 

3.03 

0.4 

3.14 

0.4 

Aug  

2.55 

0.4 

2.66 

0.4 

3.66 

0.5 

2.80 

0.4 

5.70 

0.9 

6.30 

1.0 

Sept  

6.88 

1.7 

4.04 

0.8 

2.73 

0.5 

3.38 

0.6 

3.57 

0.7 

4.49 

0.9 

Oct  

4.69 

1.7 

1.40 

0.5 

1.28 

0.4 

1.20 

0.4 

5.33 

1.2 

3.56 

0.8 

Nov  

4.70 

1.6 

2.51 

0.8 

1.75 

0.6 

3.15 

1.0 

3.06 

1.0 

2.32 

0.8 

Dec  

4.88 

2.5 

0.00 

0.5 

2.93 

1.4 

1.93 

1.1 

3.20 

1.6 

4.02 

2.0 

16.  The  present  capacity  of  the  Lake  Spaulding  reservoir  of  the  Pacific 
Gas  and  Electric  Co.  is  2,000,000,000  cu.  ft.   (it  will  eventually  be  twice 
this),  the  present  flow  is  300  cu.  ft.  per  second,  and  the  net  head  on  the 
power  house  is  approximately  1300  ft.     If  the  plant  runs  at  full  load  con- 
tinuously and  there  is  no  stream  flow  into  the  lake,  how  long  would  this 
water  last?     If  this  same  storage  capacity  were  available  for  a  plant  of  the 
same  power  under  a  head  of  40ft.,  what  rate  of  discharge  would  be  required 
and  how  long  would  the  water  last?     (It  is  worth  noting  that  the  surface 
area  of  Lake  Spaulding  is  1.3  square  miles  and  the  total  drop  in  the  water 
surface  would  be  56  feet  if  the  sides  were  vertical.     Actually  the  drop  is 
greater.     No  such  drop  in  level  would  be  found  in  connection  with  a  plant 
under  a  40-ft.  head.     If  we  assume  the  drop  in  level  to  be  10  ft.,  for  example, 
the  surface  area  of  the  storage  reservoir  would  have  to  be  233  square  miles. 
Also  the  lowering  of  the  head  on  the  plant  in  the  latter  case  would  make 
it  necessary  to  use  more  water  and  hence  shorten  the  time  as  computed.) 

Ans.     772  days,  9750  cu.  ft.  per  second,  2.4  days. 

17.  The  difference  in  elevation  between  the  surface  of  the  water  in  a 
storage  reservoir  and  the  intake  to  a  turbine  was  132.4  ft.     During  a  test 
the  pressure  at  the  latter  point  was  126.6  ft.  and  the  discharge  44.5  cu.  ft. 
per  second,  giving  a  velocity  head  in  a  30  in.  intake  of  1.3  ft.     What  was 
the  efficiency  of  the  pipe  line?     What  was  the  value  of  the  power  delivered? 

Ans.     96.6  per  cent.,  647  h.p. 

18.  Assuming  the  loss  of  head  to  be  proportional  to  the  square  of  the  rate 
of  discharge,  what  is  the  maximum  power  the  pipe  in  problem  (17)  could 
deliver?     How  many  cubic  feet  of  water  per  second  are  consumed  per 
horsepower  in  problems  (17)  and  (18)?     Ans.     1400  h.p.,  0.069,  0.088. 


WATER  POWER  29 

19.  The  pipe  line  in  problem  (17)  was  5  ft.  in  diameter.     From  the  test 
data  the  loss  of  head  may  be  computed  as 

Hf  =  5.63  V2/d  =  18,100/d5 

where  d  is  in  feet  and  the  rate  of  discharge  is  supposed  to  be  44.5  cu.  ft.  per 
second  in  every  case.  Assume  this  expression  to  be  true  for  similar  pipes 
of  different  sizes,  the  cost  of  3,  4,  5,  and  6  ft.  riveted  steel  pipes  to  be  $4.25, 
$7.50,  $12.50,  and  $18.00  per  foot  respectively,  and  the  length  of  pipe  to  be 
2000  ft.  If  the  value  of  a  horsepower  per  year  is  $20,  the  interest  and  de- 
preciation rate  7  per  cent.,  and  the  rate  of  discharge  44.5  cu.  ft.  per  second, 
what  is  the  most  economical  size  of  pipe?  Ans.  5  ft. 


CHAPTER  IV 
THE  TANGENTIAL  WATER  WHEEL 

28.  Development. — The  tangential  water  wheel  is  the  type  of 
impulse  turbine  used  in  this  country.  Its  theory  and  charac- 
teristics are  precisely  the  same  as  those  for  the  Girard  impulse 
turbine,  used  abroad,  and  the  two  differ  only  in  appearance 
and  mechanical  construction.  It  is  used  rather  than  the  Girard 
turbine,  because  of  the  advantages  offered  by  its  superior  type 


FIG.  21. — Doble  ellipsoidal  bucket. 

of  construction.  The  tangential  wheel  is  also  called  an  impulse 
wheel  or  a  Pelton  wheel  in  honor  of  the  man  who  contributed  to 
its  early  development.  The  use  of  the  term  "Pelton  water 
wheel"  does  not  necessarily  imply,  therefore,  that  it  is  the 
product  of  the  particular  company  of  that  name. 

The  development  of  this  wheel  was  begun  in  the  early  days  in 
California  but  the  present  wheel  is  a  product  of  the  last  20  years. 
For  the  purpose  of  hydraulic  mining  in  1849  numerous  water 
powers  of  fairly  high  head  were  used,  some  of  the  jets  being  as 

30 


THE  TANGENTIAL  WATER  WHEEL 


31 


much  as"2000  h.p.  When  the  gold  was  exhausted  many  of  these 
jets  were  then  used  for  power  purposes.  The  first  wheels  were 
very  crude  affairs,  often  of  wood,  with  flat  plates  upon  which 
the  water  impinged.  The  ideal  maximum  efficiency  of  a  wheel 
with  flat  vanes  is  only  50  per  cent.  The  next  improvement 
was  the  use  of  hemispherical  cups  with  the  jet  striking  them 
right  in  the  center.  A  man  by  the  name  of  Pelton  was  running 
one  of  these  wheels  one  day  when  it  came  loose  on  its  shaft  and 
slipped  over  so  that  the  water  struck  it  on  one  edge  and  was  dis- 


FIG.  22. — Allis-Chalmers  bucket.     (Courtesy  of  Allis- Chalmers  Mfg.  Co.) 

charged  from  the  other  edge.  The  wheel  was  observed  to  pick 
up  in  power  and  speed  and  this  led  to  the  development  of  the 
split  bucket. 

29.  Buckets. — The  original  type  of  Pelton  bucket  may  be  seen 
in  Fig.  76,  page  88,  the  Doble  ellipsoidal  bucket  is  shown  in 
Fig.  21,  the  Allis-Chalmers  type  in  Fig.  22,  while  the  recent 
Pelton  bucket  may  be  seen  in  Fig.  23.  In  every  case  the  jet 
strikes  the  dividing  ridge  and  is  split  into  two  halves.  The 
better  buckets  are  made  of  bronze  or  steel,  the  cheaper  ones  for 


32  HYDRAULIC  TURBINES 

low  heads  of  cast  iron.  They  are  all  polished  inside  and  the 
" splitter"  ground  to  a  knife  edge  so  as  to  reduce  friction  and 
eddy  losses  within  the  bucket.  They  may  weigh  as  much  as 
430  Ib.  apiece  and  be  from  24  to  30  in.  in  width. 

The  buckets  are  bolted  onto  a  rim.  The  interlocking  chain 
type  is  shown  in  Figs.  23  and  24.  With  this  design  each  bolt 
serves  two  buckets  in  such  a  fashion  that  the  latter  are  connected 


FIG.  23. — Pelton  bucket.     (Courtesy  ofPelton  Water  Wheel  Co.) 

as  a  chain.     The  advantage  gained  is  one  of  compactness,  it 
being  possible  to  place  the  buckets  somewhat  closer  together. 

30.  General  Proportions. — It  has  been  found  that  for  the  best 
efficiency  the  area  of  the  jet  should  not  exceed  0.1  the  projected 
area  of  the  bucket,  or  the  diameter  of  the  jet  should  not  exceed 
0.3  the  width  of  the  bucket. l  If  this  ratio  is  exceeded  the  buckets 
are  crowded  and  the  hydraulic  friction  loss  becomes  excessive. 
It  is  evident  also  that  there  must  be  some  relation  between  size 
of  jet  and  the  size  of  the  wheel.  For  a  given  size  jet  there  is  no 

i  W.  R.  Eckart,  Jr.,  Proc.  of  Inst.  of  Mech.  Eng.  (London),  Jan.  7,  1910. 


THE  TANGENTIAL  WATER  WHEEL  33 

upper  limit  as  to  size  of  wheel  so  far  as  the  hydraulics  is  concerned. 
In  special  cases,  where  a  low  r.p.m.  was  desired,  diameters  as 
large  as  35  ft.  have  been  used  when  the  diameter  of  the  jet  was 
only  a  few  inches.  But  there  is  a  lower  limit  for  the  ratio  of 
wheel  diameter  to  jet  diameter.  Obviously,  for  instance,  the 
wheel  could  not  be  as  small  as  the  jet.  The  considerations  which 
influence  this  matter  will  be  further  considered  in  Chapter  VII, 
but  for  the  present  it  will  be  sufficient  to  state  that  a' ratio  as 
low  as  9  may  be  used  without  an  excessive  loss  of  efficiency.1 
(The  nominal  diameter  is  that  of  a  circle  tangent  to  the  center 


FIG.  24. — Pelton   tangential  water  wheel  runner  showing  interlocking   chain- 
type  construction.     (Made  by  Pelton  Water  Wheel  Co.) 

line  of  the  jet.)  The  more  common  value,  and  one  which 
involves  no  sacrifice  of  efficiency,  is  12.  From  that  we  get  a 
very  convenient  rule  that  the  diameter  of  the  wheel  in  feet  equals 
the  diameter  of  the  jet  in  inches.  The  size  of  jet  necessary  to 
develop  a  given  amount  of  power  under  any  head  may  be  com- 
puted and  then  the  diameter  of  wheel  necessary  is  known  at  once. 
The  r.p.m.  of  the  wheel  can  be  computed  by  taking  the  periph- 
eral speed  as  0.47  of  the  jet  velocity  or  0.45  -\/2gh. 

XS.  J.   Zowski,    "Water  Turbines,"  published  by  Engineering  Society, 
Univ.  of  Mich.,  1910. 
3 


34 


HYDRAULIC  TURBINES 


The  use  of  one  jet  only  upon  a  single  wheel  is  to  be  preferred 
if  it  is  possible.  However,  two  jets  are  often  used  upon  one 
wheel  though  at  some  sacrifice  of  efficiency.  For  a  given  size 
wheel  the  horsepower  of  one  jet  is  limited  by  the  maximum  size 
of  the  jet  that  may  be  employed.  If  a  greater  horsepower  is 
desired  it  is  necessary  to  use  two  or  more  jets  upon  the  one 
wheel  or  to  use  a  larger  wheel  with  a  single  jet.  The  larger  whool 


FIG.  25. — Tangential  water  wheel  unit  with  deflecting  nozzle. 

means  a  lower  r.p.m.  and  a  higher  cost  both  of  the  wheel  and  the 
generator  if  a  direct  connected  unit  is  used.  In  case  this  addi- 
tional expense  is  not  justified  by  the  increased  efficiency  of  the 
single  jet  wheel  the  duplex  nozzle  would  be  used. 

The  tangential  water  wheel  is  almost  always  set  with  a  hori- 
zontal shaft  and,  if  direct  connected  to  a  generator,  is  overhung 
so  that  the  unit  has  only  two  bearings  (Fig.  25).  It  is  quite 
common  for  two  wheels  to  drive  a  single  generator  mounted 
between  them  in  which  case  we  have  the  double-overhung  type. 


THE  TANGENTIAL  WATER  WHEEL 


35 


31.  Speed  Regulation. — Various  means  have  been  adopted  to 
regulate  the  power  input  to  the  tangential  water  wheel  but  the 
following  are  the  only  ones  that  are  of  any  importance.  The 


FIG.  26.  —  5286   h.p.    Jet,   from 
velocity 


in-    needle   nozzle.     Head  =  822   ft.     Jet 
227.4  ft.  per  second. 


use  of  any  throttle  valve  in  the  pipe  line  is  wasteful  as  it  destroys 
a  portion  of  the  available  head  and  thus  requires  more  water  to 
be  used  for  a  given  amount  of  power  than  would  otherwise  be  the 
case.  The  ideal  mode  of  governing,  so  far  as  economy  of  water 


FIG.  27. — Deflecting  needle  nozzle.     (After  drawing  by  Prof.  W.  R.  Eckart,  Jr.) 

is  concerned,  would  not  affect  the  head  but  would  merely  vary 
the  water  used  in  direct  proportion  to  the  power  demanded. 
The  needle  nozzle  (Fig.  27)  accomplishes  this  result  very  nearly. 


36  HYDRAULIC  TURBINES 

As  the  needle  is  moved  back  and  forth  it  varies  the  area  of  the 
opening  and  thus  varies  the  amount  of  water  discharged.  The 
coefficient  of  velocity  is  a  maximum  when  the  nozzle  is  wide 
open  but  it  does  not  decrease  very  seriously  for  the  smaller-nozzle 
openings.  (See  Fig.  89.)  Thus  the  velocity  of  the  jet  is  very 
nearly  the  same  for  all  values  of  discharge.  The  efficiency  of  a 
well-constructed  needle  nozzle  is  very  high,  being  from  95  to  98 
per  cent.1  The  needle  nozzle  is  nearly  ideal  for  economy  of  water 
but  may  not  always  permit  close  speed  regulation.  If  the  pipe 
line  is  not  too  long,  the  velocity  of  flow  low,  and  the  changes  of 
load  small  and  gradual,  the  needle  nozzle  may  be  very  satis- 
factory. In  case  it  is  used  the  penstock  is  usually  provided  with 
&_standpipe  or  a  surge  tank. 

If  the  pipe  line  is  long,  the  velocity  of  flow  high,  and  the  changes 
of  load  severe,  dangerous  water  hammer  might  be  set  up  if  the 
discharge  were  changed  too  quickly.  It  fought  therefore  be 
difficult  to  secure  close  speed  regulation  with  the  needier  nozzle 
as  the  governors  would  have  to  act  slowly.  The  deflecting  nozzle, 
shown  in  Fig.  7,  page  8,  is  much  used  for  such  cases.  The 
nozzle  is  made  with  a  ball-and-socket  joint  so  that  the  entire  jet 
can  be  deflected  below  the  wheel  if  necessary.  The  governor 
sets  the  nozzle  in  such  a  position  that  just  enough  water  strikes 
the  buckets  to  supply  the  power  demanded.  The  rest  of  the 
water  passes  below  the  buckets  and  is  wasted.  Since  there  is  no 
change  in  the  flow  in  the  pipe  line  the  governor  may  accomplish 
any  degree  of  speed  regulation  desired  as  there  is  little  limit  to 
the  rapidity  with  which  the  jet  may  be  deflected.  Such  a  nozzle 
is  usually  provided  with  a  needle  also  which  is  regulated  by  hand. 
Fig.  27  is  really  of  this  type.  In  another  type  the  body  of  the 
nozzle  is  stationary  and  only  the  tip  is  moved.  The  needle 
stem  must  be  equipped  with  a  guide  in  this  moving  part  and 
also  be  fitted  with  a  universal  joint  so  that  the  needle  point  may 
always  remain  in  the  center  of  the  jet.  The  station  attendant 
sets  the  needle  from  time  to  time  according  to  the  load  that  he 
expects  to  carry.  However,  the  device  is  wasteful  of  water 
unless  carefully  watched.  If  other  water  rights  prevent  the  flow 
of  a  stream  from  being  interfered  with  it  may  be  satisfactory. 

In  some  modern  plants  the  operator  can  control  the  position 
of  the  needle  from  the  switchboard  and  by  careful  attention  very 

!W.  R.  Eckart,  Jr.,  Inst.  of  Mech.  Eng.  (London),  Jan.  7,  1910. 
Bulletin  No.  6,  Abner  Doble  Co. 


THE  TANGENTIAL  WATER  WHEEL 


37 


little  water  is  wasted.  Since  the  experience  is  that  loads  in- 
crease slowly,  the  operator  need  have  little  trouble  in  keeping 
the  unit  up  to  speed. 

The  combined  needle  and  deflecting  nozzle  may  possess  the 
advantages  of  both  of  the  above  types,  by  having  the  needle 
automatically  operated.  If  the  load  on  the  wheel  is  reduced 


M  \  EIG.  28. — Deflecting  needle  nozzle  for  8000  h.p.  wheel. 

the  goveriftfr  at  once  deflects  the  jet  thus  preventing  any  increase 
of  speed.  .Then  a  secondary  relay  device  slowly  closes  the  needle 
nozzle  and,jas  it  does  so,  the  nozzle  is  gradually  brought  back  to 
its  original  "position  where  all  the  water  is  used  upon  the  wheel. 
Thus  close  speed  regulation  is  accomplished  with  very  little 
waste  of  water. 


FIG.  29.— Needles  and  nozzle  tips.      (Courtesy  ofPelton  Water  Wheel  Co.) 

The  needle  nozzle  with  auxiliary  relief  shown  in  Fig.  30  and  Fig. 
31  accomplishes  the  same  results  as  the  above.  When  the  needle 
of  the  main  nozzle  is  closed  the  auxiliary  nozzle  underneath  it  is 
opened  at  the  same  time.  This  discharges  an  equivalent  amount 
of  water  which  does  notjstrike  the  wheel.  This  auxiliary  nozzle 
is  then  slowly  closed  by  means  of  a  dash-pot  mechanism.  While 


38 


i    HYDRAULIC  TURBINES 


both  of  these  types  relieve  the  pressure  in  case  of  a  decreasing 
load  they  are  unable  to  afford  any  assistance  in  the  case  of  a 
rapid  demand  for  water.  The  deflecting  nozzle  alone  is  the  only 
type  that  is  perfect  there. 

32.  Conditions  of  Use. — The  tangential  water  wheel  is  best 
adapted  for  high  heads  and  relatively  small  quantities  of  water. 
By  that  is  meant  that  the  choice  of  the  type  of  turbine  is  a  func- 
tion of  the  capacity  as  well  as  the  head.  For  a  given  head  the 
larger  the  horsepower,  the  less  reason  there  is  for  using  this 
type  of  wheel. 


FIG.  30. — Auxiliary  relief  needle  nozzle. 
(Made  by  Pelton  Water  Wheel  Co.) 

In  Switzerland  a  head  as  high  as  5412  ft.  has  been  used  for 
5  wheels  of  3000  h.p.  each.  The  jets  are  1.5  in.  in  diameter  and 
the  wheels,  which  run  at  500  r.p.m.,  are  11.5  ft.  in  diameter. 
There  are  several  installations  in  this  country  under  heads  of 
about  2100  ft.  There  are  numerous  cases  of  heads  between 
1000  and  2000  ft.  but  probably  the  majority  of  the  installations 
are  for  heads  of  about  1000  ft. 

The  largest  power  developed  by  a  single  jet  upon  a  single 
wheel  is  15,000  h.p.  The  jet  is  8  in.  in  diameter  and  the  wheel 
runs  at  375  r.p.m.  under  a  head  of  about  1700  ft. 


THE  TANGENTIAL  WATER  WHEEL  39 

The  largest  jet  employed  upon  any  Pelton  wheel  is  about 
in.  in  diameter.  The  net  head  is  506  ft.  in  this  case. 
There  are  a  number  of  large  jets  of  9  in.  or  over  used  for  heads 
trom  900  to  1500  ft. 

33.  Efficiency. — The  efficiency  of  the  tangential  water  wheel 
is  about  the  same  as  that  of  the  average  reaction  turbine.  From 
75  to  85  per  cent,  may  reasonably  be  expected  though  lower 
yafiSfare  6FtelTolDtamed7~due  to  poor  design. 


FIG.  31.— Auxiliary  relief  needle  nozzle  for  use  with  10,000  kw.  tangential  water 
wheel.     (Made  by  Pelton  Water  Wheel  Co.) 


34.  QUESTIONS  AND  PROBLEMS 

1.  Of  what  materials  are  impulse  wheel  buckets  constructed?     How  are 
they  secured  to  the  rim?     What  is  the  advantage  of  "chain  type"  con- 
struction?    What  should  be  the  relation  between  the  size  of  the  jet  and  the 
size  of  the  bucket? 

2.  When  would  two  or  more  jets  be  used  upon  a  Pelton  wheel?     What 
is  the  relation  between  the  diameter  of  the  wheel  and  the  diameter  of  the 
jet?     How  may  the  speed  of  rotation  of  a  wheel  of  given  diameter  be  com- 
puted, if  the  head  is  known?     What  fixes  the  diameter  of  the  jet  that  is 
to  be  employed,  assuming  that  it  is  not  limited  by  any  wheel  size? 

3.  What  is  meant  by  single-overhung  and  double-overhung  construction? 
What  is  the  advantage  of  the  latter?     How  is  the  shaft  usually  placed? 

4.  What  is  the  needle  nozzle,  the  deflecting  nozzle,  and  the  deflecting 


40  HYDRAULIC  TURBINES 

needle  nozzle?     What  is  the  needle  nozzle  with  auxiliary  relief  and  how 
does  it  operate? 

5.  What  are  the  relative  merits  of  the  different  methods  of  governing 
the  tangential  water  wheel? 

6.  What  are  the  conditions  of  use  of  impulse  wheels  in  regard  to  head, 
power,  size  of  jet,  etc.?     What  efficiency  should  be  expected? 

7.  It  is  desired  to  develop  3880  h.p.  with  a  Pelton  wheel  under  a  head 
of  900  ft.    Assuming  the  efficiency  of  the  wheel  to  be  82  per  cent,  and  the 
velocity  coefficient  of  the  nozzle  to  be  0.98,  what  will  be  the  diameter  of 
the  jet?     What  will  then  be  a  reasonable  diameter  for  the  wheel  and  its 
probable  speed  of  rotation?  Ans.     6  in.,  6  ft.,  345  rev.  per  min. 

8.  How  small  could  the  wheel  be  made  in  the  preceding  problem  ?     What 
would  then  be  its  speed  of  rotation  ?     If  a  higher  speed  than  this  is  desired 
for  the  same  horsepower,  what  construction  could  be  employed? 

9.  A  Pelton  wheel  runs  at  a  constant  speed  under  a  head  of  625  ft.     The 
cross-section  area  of  the  jet  is  0.200  sq.  ft.  and  the  nozzle  friction  loss  is  to 
be  neglected.     Suppose  a  throttle  valve  in  the  pipe  reduces  the  head  at 
the  base  of  the  nozzle  from  625  ft.  to  400  ft.     Under  these  conditions  the 
efficiency  of  the  wheel  (the  speed  of  the  wheel  no  longer  being  proper  for  the 
head)  is  known  to  be  50  per  cent.     Find  the  rate  of  discharge,  power  of  jet, 
and  power  output  of  wheel. 

Ans.     32.08  cu.  ft.  per  second,  1458  h.p.,  729  h.p. 

10.  A  Pelton  wheel  runs  at  a  constant  speed   under  a  head  of  625  ft. 
The  cross-section  area  of  the  jet  is  0.200  sq.  ft.  and  the  nozzle  friction  loss 
is  to  be  neglected.     Suppose  the  needle  of  the  nozzle  is  so  adjusted  as  to 
reduce  the  area  of  the  jet  from  0.200  to  0.0732  sq.  ft.     Under  these  condi- 
tions the  efficiency  of  the  wheel  is  known  to  be  70  per  cent.     Find  the  rate 
of  discharge,  power  of  jet,  and  power  output  of  wheel. 

Ans.  14.67  cu.  ft.  per  second,  1041  h.p.,  729  h.p. 

11.  Compare  the  water  consumed  per  horsepower  output  for  the  wheel 
jn  the  preceding  two  problems.     Compute  the  overall   efficiency  in  each 
case  using  the  head  of  625  ft.  Ans.     32  per  cent.,  70  per  cent. 


CHAPTER  V 
THE  REACTION  TURBINE 

35.  Development. — The  primitive  type  of  reaction  turbine 
known  as  Barker's  Mill  is  shown  in  Fig.  32.  The  reaction  of  the 
jets  of  water  from  the  orifices  causes  the  device  to  rotate.  In 
order  to  improve  the  conditions  of  flow  the  arms  were  then  curved 
and  it  became  known  in  this  form  as  the  Scotch  turbine.  Then 
three  or  more  arms  were  used  in  order  to  increase  the  power,  and 
with  still  further  demands  for  power  more  arms  were  added  and 
the  orifices  made  somewhat  larger  until  the  final  result  was  a 
complete  wheel.  In  1826  a  French 
engineer,  Fourneyron,  placed  station- 
ary guide  vanes  within  the  center  to 
direct  the  water  as  it  flowed  into  the 
wheel  and  we  then  had  the  outward 
flow  turbine.  In  1843  the  first  Four- 
neyron turbines  were  built  in  America.  x  ^ T-, 

The  axial  flow  turbine  commonly    _j_ (^  j)  I 

called  the  Jonval  was  also  a  Euro- 
pean   design    introduced  .  into    this        ,., 

FIG.  32. — Barker  s  mill. 

country  in  1850. 

An  inward  flow  turbine  was  proposed  by  Poncelot  in  1826  but 
the  first  one  was  actually  built  by  Howd,  of  New  York,  in  1838. 
The  latter  obtained  a  patent  and  installed  several  wheels  of  crude 
workmanship  in  the  New  England  mills.  In  1849  James  B. 
Francis  designed  a  turbine  under  this  patent  but  his  wheel  was 
of  superior  construction.  Furthermore  he  conducted  accurate 
tests,  published  the  results,  analyzed  them,  and  formulated  rules 
for  turbine  runner  design.  He  thus  brought  this  type  of  wheel 
to  the  attention  of  the  engineering  world  and  hence  his  name 
became  attached  to  it. 

The  original  Francis  turbine  is  shown  in  Fig.  5,  page  4,  and 
in  Fig.  6,  page  5,  may  be  seen  photographs  of  a  radial  inward 
flow  runner  of  this  type  though  of  more  recent  date.  As  may  be 
seen  in  Fig.  4,  page  3,  the  water  has  to  turn  and  flow  away 

41 


42 


HYDRAULIC  TURBINES 


axially  after  its  discharge  and  hence  the  original  design  was 
gradually  modified  so  that  the  water  began  to  turn  before  its 
discharge  from  the  runner.  The  Swain  turbine  (1855)  shows 
this  evolution  and  the  McCormick  runner  (1876)  carries  it  still 
further.  The  latter  is  the  prototype  of  the  modern  high  speed 
mixed  flow  runner.  The  nearest  approach  to  the  original  Francis 
runner  in  present  practice  is  to  be  seen  in  Fig.  34,  Type  I,  and  in 
Fig.  36.  The  pure  radial  flow  turbine  is  no  longer  built,  but 
since  all  the  modern  inward  mixed  flow  turbines  may  be  said  to 
have  grown  out  of  it,  they  are  today  quite  generally  known  as 
Francis  turbines. 


Howd     1838 


Francis    1849 


Swain     1855 


FIG.  33.- 


McCormick    1876 
-Evolution  of  the  modern  turbine. 


The  high-speed  mixed  flow  runner,  illustrated  by  the  original 
McCormick  type  in  Fig.  33,  arose  as  the  result  of  a  demand  for 
higher  speed  and  power  under  the  low  falls  first  used  in  this 
country.  Higher  speed  of  rotation  could  be  obtained  by  using 
runners  of  smaller  diameter,  but  higher  power  required  runners 
of  larger  diameter,  so  long  as  the  same  designs  were  adhered  to. 
So  in  order  to  increase  the  capacity  of  a  wheel  of  the  same  or 
smaller  diameter,  the  design  was  altered  by  making  the  depth 
of  the  runner  greater  (i.e.,  the  dimension  B,  Fig.  34,  was  in- 
creased). The  area  of  the  waterway  through  the  runner  was 


THE  REACTION  TURBINE 


43 


44 


HYDRAULIC  TURBINES 


also  increased  slightly  by  using  fewer  vanes  and  it  was  then  de- 
sirable to  extend  these  further  in  toward  the  center.  As  that 
left  a  very  small  space  in  the  center  for  the  water  to  discharge 
through,  it  was  necessary  for  the  runner  to  discharge  the  greater 
part  of  the  water  axially.  Type  IV  of  Fig.  34  shows  the  high- 
speed high-capacity  runner  of  today. 


FIG.  35. — Leffel  turbine  for  open  flume.     (Made  by  James  Leffel  and  Co.) 

As  civilization  moved  from  the  valleys,  where  the  low  falls  were 
found,  up  into  the  more  mountainous  regions,  and  as  means  of 
transmitting  power  were  introduced,  it  became  desirable  to 
develop  higher  heads,  and  in  1890  a  demand  arose  for  high- 
head  wheels  which  American  builders  were  not  able  to  supply. 
For  a  time  European  designs  were  used  and  then  it  was  seen  that 


THE  REACTION  TURBINE  45 

a  type  similar  to  the  original  Francis  turbine  was  well  suited  to 
those  conditions.     This  is  shown  by  Type  I  of  Fig.  34. 


FIG.  36. — 42"  Francis  runner.     8000  h.p.,  600  ft.  head. 
(Made  by  Platt  Iron  Works  Co.) 


FIG.  37. — Turbine  runners  of  the  Allis-Chalmers  Mfg.  Co. 

At  present  the  range  of  common  American  practice  is  covered 
by  the  four  types  shown  in  Fig.  34,  though  in  a  few  cases  extreme 


46  HYDRAULIC  TURBINES 

designs  have  passed  beyond  these  limits.  American  turbines 
in  the  past  were  developed  by  "cut  and  try"  methods,  European 
turbines  largely  by  mathematical  analysis.  At  the  present 
time  the  best  turbines  in  this  country  are  designed  from  rational 
theory  supplemented  by  experimental  investigation. 

36.  Advantages  of  Inward  Flow  Turbine. — The  Fourneyron 
turbine  has  a  high  efficiency  on  full  load  and  is  useful  in  some  cases 
where  a  low  speed  is  desired,  but  it  has  been  supplanted  by  the 
Francis  turbine  for  the  following  reasons: 

1.  The  inward  flow  turbine  is  much  more  compact,  the  runner 
can  be  cast  in  one  piece,  and  the  whole  construction  is  better 
mechanically. 


(From  a  photograph  by  the  author.) 
FIG.  38. — Construction  of  a  built-up  runner. 

2.  Since  the  turbine  is  more  compact  and  smaller,  the  con- 
struction will  be  much  cheaper.     The  smaller  runner  will  permit 
of  a  higher  r.p.m.  and  that  means  a  cheaper  generator  can  be 
used. 

3.  The  gates  for  governing  are  more  accessible  and  it  is  easier 
to  construct  them  so  as  to  minimize  the  losses.     Thus  the  effi- 
ciency of  the  turbine  on  part  load  is  better  than  is  the  case  with 
the  outward  flow  type. 

4.  It  is  easier  to  secure  the  converging  passages  that  are  neces- 
sary through  the  runner. 

5.  A  draft  tube  can  be  more  conveniently  and  effectively  used. 


THE  REACTION  TURBINE 


47 


37.  General  Proportions  of  Types  of  Runners. — It  has  already 
been  seen  how  the  need  for  varying  the  capacity  of  runners 
without  changing  their  diameters  has  led  to  altering  the  height 
n  and  the  general  profile  as  illustrated  in  Fig.  34.  The  increased 
volume  of  water  through  the  higher  capacity  runners  also  re- 
quires a  larger  diameter  of  draft  tube,  as  well  as  a  higher  velocity 


mm. 

(From  a  photograph  by  the  author.) 
FIG.  39. — Double-discharge  runner. 

of  flow  at  this  section,  and  in  extreme  types  the  flow  through  the 
runner  is  not  merely  inward  and  downward  but  for  those  particles 
of  water  nearest  the  band  or  ring  it  is  inward,  downward,  and 
outward. 

But  the  quantity  of  water  which  will  flow  through  the  runner 
depends  not  only  upon  the  area  at  inlet  but  also  upon  the  velocity 
of  the  water.  If  we  confine  our  attention  to  the  circumferential 
area  of  the  runner  at  entrance  we  are  concerned  with  the  velocity 
normal  to  it  and  this  is  the  radial  component  of  velocity.  Hence 
we  increase  the  capacity  of  the  runner  by  making  the  radial 


48  HYDRAULIC  TURBINES 

component  of  the  velocity  of  the  water  larger.  This  causes  the 
angle  a i  to  be  increased  as  may  be  seen  in  Fig.  34.  The  angle  a\ 
is  determined  by  the  guide  vanes. 

It  is  convenient  to  express  the  peripheral  velocity  of  the  runner 
HI  as  equal  to  <t>\/2gh.  The  value  of  0  which  gives  the  most 
efficient  speed  for  a  given  turbine  is  denoted  by  <f>e  and  values  of 
<f>e  for  different  turbines  range  from  about  0.55  to  about  0.90 
according,  to  the  design.1  If  the  value  of  <£  is  higher  than  this 
it  is  probable  that  the  speed  is  higher  than  the  best  speed  or 
that  the  nominal  diameter  for  which  u\  is  computed  is  larger  than 
the  real  diameter.  Values  of  4>e  may  be  varied  in  the  design  by 
altering  certain  angles  and  areas  of  the  runner. 

Since  it  is  desirable,  in  general,  to  increase  or  decrease  the 
rotative  speed  and  the  capacity  simultaneously,  the  custom  is 
to  so  proportion  the  runners  that  low  values  of  (f>e  are  found  with 
turbines  of  Type  I,  Fig.  34,  while  high  values  are  found  with 
those  of  Type  IV.  Thus  a  low-capacity  runner  also  has  a  low 
peripheral  speed  for  a  given  head,  while  a  high-capacity  runner 
would  have  a  higher  peripheral  speed.  Thus  for  a  given  diameter 
of  runner  under  a  given  head  both  power  and  speed  of  rotation 
increase  from  Type  I  to  Type  IV.  If,  on  the  other  hand,  the 
power  is  fixed,  the  diameter  of  runner  of  Type  IV  would  be 
much  smaller  than  that  of  Type  I.  Hence  the  rotative  speed  of 
the  former  would  be  higher  due  to  the  smaller  diameter  as  well 
as  the  increased  linear  velocity.  For  this  reason  this  type  is 
called  a  high-speed  runner,  while  Type  I  is  a  low-speed  runner. 
Both  capacity  and  speed  are  involved  in  a  single  factor  variously 
known  as  the  specific  speed,  characteristic  speed,  unit  speed,  and 
type  characteristic.  It  is  N8  =  Ne\/B.hp./h5/4:,  the  derivation  of 
which  will  be  given  later.  (Ne  is  the  speed  for  highest  efficiency.) 
As  the  capacity  and  speed  increase,  this  factor  increases.  Hence 
a  " high-speed"  turbine  is  really  a  high  specific  speed  turbine  and 
a  "low-speed"  turbine  is  a  low  specific  speed  turbine.  The 
value  of  N,  is  an  index  of  the  type  of  turbine.  Values  of  N8  for 
reaction  turbines  range  from  10  to  100,  though  the  latter  limit 
is  occasionally  exceeded. 

The  vector  diagrams  of  the  velocities  at  entrance  are  drawn  to 
the  same  scale  in  Fig.  34  as  if  all  four  types  were  under  the  same 
head.  It  may  be  seen  that  as  we  proceed  from  Type  I  to  Type 
IV,  Ui,  ai,  and  /3'i  increase,  while  V\  decreases.  Since  the  angle 

1  "Water  Turbines,"  by  S.  J.  Zowski  is  the  source  of  much  of  Fig.  34. 


THE  REACTION  TURBINE 


49 


0i  is  the  angle  which  the  relative  velocity  of  the  water  makes  at 
entrance,  the  vane  angle  fi'i  should  be  made  equal  to  it. 

38.  Comparison  of  Types  of  Runners. — As  a  means  of  illustrat- 
ing the  differences  between  the  various  types  of  runners  the 
following  tables  are  presented: 


Ring 


FIG.  40. — Methods  of  specifying  runner  diameter. 
TABLE  1. — COMPARISON  OF  12-iN.  WHEELS  UNDER  30-FT.   HEAD 


Type 

Discharge, 
cu.  ft.  per 

minute 

H.p. 

R.p.m. 

Tangential  water  wheel  

7.9 

0  37 

380 

Reaction  turbines: 
Tvne  I  . 

99.0 

4.3 

460 

Type  II 

329  0 

14  9 

554 

Type  III 

741  0 

33  4 

600 

Type  IV 

1209.0 

55  5 

730 

TABLE  2. — COMPARISON  OF  WHEELS  TO  DEVELOP  15  H.p.  UNDER 
30-FT.  HEAD 


Type 

Diameter,  in. 

R.p.m. 

Tangential  water  wheel  

60 

1    55 

Reaction  turbines: 
Tvne  I 

21 

274 

Type  II 

12 

554 

Type  III                                         .            . 

8 

900 

Type  IV.. 

6 

1460 

It  will  be  seen  that  the  tangential  water  wheel  is  a  low-speed, 
low-capacity  type,  while  the  reaction  turbine  of  Type  IV  is  a  high- 
speed, high-capacity  runner.  This  may  be  contrary  to  the  popu- 
lar impression,  but  these  terms  as  used  here  have  only  relative 


50 


HYDRAULIC  TURBINES 


meanings.  Under  high  heads  where  the  r.p.m.  would  naturally 
be  high  the  relatively  lower  speed  of  the  tangential  water  wheel 
is  of  advantage,  while  under  the  low  heads  the  relatively  higher 
speed  of  the  reaction  turbine  is  of  advantage.  This  difference  of 
speed  exists  even  when  the  runners  are  of  the  same  diameter  as 
seen  by  the  first  table.  But  when  the  diameters  are  made  such  as 
to  give  the  same  power  as  in  the  second  table  the  difference  be- 


FIG.  41. — 13,500  h.p.  runner.     Head  =  53  ft.,   speed 

I.  P.  Morris  Co.) 


94  r.p.m.     (Made   by 


comes  much  greater.  It  must  be  understood  that  these  tables  do 
not  prove  one  type  of  wheel  to  be  any  better  than  another  but 
merely  show  what  may  be  obtained.  If  the  tangential  water 
wheel  or  Type  I  of  the  reaction  turbines  appear  in  an  unfavorable 
light  it  is  only  because  the  head  and  horsepower  are  not  suitable 
for  them. 


THE  REACTION  TURBINE 


51 


39.  Runners. — Runners  may  be  cast  solid  or  built  up,  but  the 
majority  are  cast  solid  as  the  construction  is  more  substantial. 
Occasionally  a  very  large  runner  may  be  cast  in  sections.  Built 
up  wheels  have  the  vanes  shaped  from  steel  plates  andfthe 
crown,  hubs,  and  rings  are  cast  to  them,  as  shown  in  Fig.*  38. 
The  best  runners  are  made  of  bronze.  Cast  steel  is  used  for  very 


FIG.  42. — 10,000   h.p.   runner   at   Keokuk,    la.     Head  =  32   ft.,    speed  =  57.7 
r.p.m.     (Made  by  Wellman-Seaver-M organ  Co.) 

high  heads  in  some  cases,  while  cheaper  runners  are  made  of 
cast  iron.  Very  naturally  the  large  runners  are  made  of  the 
latter  metal. 

Runners  may  be  divided  into  two  broad  classes  of  single  and 
double  discharge  runners.  Figs.  36  and  42  are  of  the  first  type 
and  Fig.  39  of  the  second.  The  latter  is  essentially  two  single 


52 


HYDRAULIC  TURBINES 


discharge  runners  placed  back  to  back  and  requires  two  draft 
tubes  as  the  water  is  discharged  from  both  sides.  It  is  used 
only  for  horizontal  shaft  units,  while  the  single  discharge  runner 
may  be  used  for  either  horizontal  or  vertical  shaft  turbines. 

Turbines  are  often  rated  according  to  the  diameter  of  the 
runner  in  inches.  This  diameter  is  easily  fixed  in  many  cases, 
but  for  one  of  the  type  shown  in  Fig.  40  either  one  of  four  dimen- 
sions may  be  used.  Different  makers  follow  different  practices 
in  this  regard  but  the  usual  method  is  to  measure  the  diameter 
at  a  point  about  halfway  down  the  entrance  height. 


Position  of  Gate 
when  Closed 


No.  1  Runner 
No.  2  Chute  Case 
No.  3  Gate 


FIG.  43. — Register  gate. 


40.  Speed  Regulation. — The  amount  of  water  supplied  to  the 
reaction  turbine  is  regulated  by  means  of  gates  of  which  there  are 
three  types. 

The  cylinder  gate  is  shown  in  Fig.  5,  page  4.  It  is  the  simplest 
and  cheapest  form  of  gate  and  also  the  poorest,  although,  when 
closed,  it  will  not  leak  as  badly  as  the  others.  When  the  gate 
is  partially  closed  there  is  a  big  shock  loss  in  the  water  entering 
the  turbine  runner  due  to  the  sudden  contraction  and  the  sudden 
expansion  of  the  stream  that  must  take  place.  With  this  type  of 


THE  REACTION  TURBINE 


53 


gate  the  efficiency  on  part  load  is  relatively  low  and  the  maximum 
efficiency  is  obtained  when  the  gate  is  completely  raised. 

A  better  type  of  gate  is  the  register  gate  shown  in  Fig.  43.  With 
this  type  the  guide  vanes  are  made  in  two  parts,  the  inner  portion 
next  to  the  runner  is  stationary,  the  outer  portion  is  on  a  ring 
which  may  be  rotated  far  enough  to  shut  the  water  off  entirely, 
if  necessary,  as  shown  by  the  dotted  lines.  While  this  is  more 


(Courtesy  of  Allis-Chalmers  Co.) 
FIG.  44. — View  of  guide  vanes  and  shifting  ring. 


efficient  than  the  preceding  type  there  is  still  a  certain  amount  of 
eddy  loss  that  cannot  be  avoided.     It  is  seldom  used. 

The  wicket  gate,  also  called  the  swing  gate  or  the  pivoted 
guide  vane,  is  shown  in  Fig.  45.  This  is  the  best  type  and  also 
the  most  expensive.  As  the  vanes  are  rotated  about  their  pivots 
the  area  of  the  passages  through  them  is  altered.  The  vanes 
may  be  closed  up  so  as  to  shut  off  the  water  if  necessary.  Of 
course  the  angle,  «i,  is  altered  and  a  certain  amount  of  eddy  loss 
may  also  result  but  it  is  less  than  occasioned  by  either  of  the 


54 


HYDRAULIC  TURBINES 


FIG.  45. — AA'icket  gate  with  all  operating  parts  outside. 


FIG.  46. — Wicket  gates  and  runner  in  turbine  made  by  Platt  Ironworks 


THE  REACTION  TURBINE 


55 


other  forms.     The  maximum  efficiency  is  obtained  before  the 
gates  are  opened  to  the  greatest  extent. 

The  connecting  rod  from  the  relay  governor  operates  a  shifting 
ring.     This  in  turn,  by  means  of  links,  rotates  the  vanes.     These 


FIG.  47.— 10,000  h.p.  turbine  at  Keokuk,  la. 

Co.) 


(Made  by  Wellman-Seaver-M  organ 


links  are  shown  in  Figs.  45,  47,  and  48.  Often  the  shifting  ring 
and  links  are  inside  the  case,  but  the  better,  though  more  ex- 
pensive, type  has  the  working  parts  outside  the  case. 

In  order  to  prevent  shock  in  the  penstock  when  the  governor 
quickly   closes   the   gates,    many   turbines  .are   provided   with 


56  HYDRAULIC  TURBINES 

mechanically  operated  relief  valves,  as  in  the  left  hand  side  of 
Fig.  49.  This  valve  is  opened  at  the  same  time  the  gates  are 
closed,  thus  by-passing  the  water.  The  relief  valve  may  be  so 
arranged  with  a  dash-pot  mechanism  that  it  will  slowly  close. 

41.  Bearings. — For  small  vertical  shaft  turbines  a  step  bearing 
made  of  lignum  vitse  is  used  under  water,  as  at  the  bottom  of 
the  runner  in  Fig.  35.  This  wood  gives  good  results  for  such 
service  and  wears  reasonably  well.  For  larger  turbines  a  thrust 
bearing  is  usually  provided  to  which  oil  is  supplied  under  pressure. 
Roller  bearings  are  also  used  with  the  rollers  running  in  an  oil 
bath,  as  in  Fig.  50.  Sometimes  rollers  are  provided  in  the  former 
type  but  act  only  when  the  pressure  fails,  and  again  roller  bear- 


FIG.  48. — Shifting  ring  and  links  on  a  wicket  gate  spiral  case  turbine.     (Made 
by  Platt  Iron  Works  Co.) 

ings  may  sometimes  be  supplied  with  oil  under  pressure  between 
two  bearing  surfaces  in  case  the  rollers  fail.  The  Kingsbury 
bearing  is  fitted  with  a  number  of  metal  shoes  so  mounted  that 
their  bearing  surfaces  are  not  quite  level.  Thus  as  they  advance 
through  the  oil  bath  a  wedge-shaped  film  of  oil  is  forced  in  be- 
tween these  shoes  and  the  other  surface.  Such  a  bearing  is 
preferably  located  at  the  top  of  the  shaft  in  which  case  it  is 
called  a  suspension  bearing,  though  it  may  be  placed  between  the 
generator  and  the  runner. 

A  horizontal  turbine  set  in  an  open  flume  often  has  lignum 
vitae  bearings  as  the  water  is  a  sufficient  lubricant.     However 


THE  REACTION  TURBINE 


57 


(Courtesy  of  Pelt  on  Water  Wheel  Co.) 
Fie;.  49. —  Spiral  case  turbine  with  relief  valve. 


(Courtesy  of  Electric  Machinery  Co.) 
FIG.  50. — Heavy  duty  suspension  bearing.  | 


58 


HYDRAULIC  TURBINES 


the  water  must  be  clear;  gritty  water  would  destroy  the  bearings. 
If  the  turbine  is  in  a  case  so  that  the  bearings  are  accessible  the 
usual  types  of  bearings  are  used.  It  must  not  be  forgotten  that 
even  though  the  shaft  be  horizontal  a  very  considerable  end  thrust 
must  be  allowed  for  due  to  the  reaction  of  the  streams  discharged 
from  the  runner.  That  is  one  reason  for  using  runners  in  pairs. 
Also  a  single  runner  is  often  used  which  has  a  double  discharge. 
(See  Fig.  51.)  Single  discharge  runners  are  often  provided  with 
some  form  of  automatic  hydraulic  balancing  piston  to  equalize 
the  thrust. 


FIG.  51. — Double  discharge  runner  in  spiral  case. 

As  the  leakage  of  water  through  the  gates,  when  closed,  may 
be  sufficient  to  keep  the  turbine  running  slowly  under  no-load, 
large  units  are  often  provided  with  brakes  so  they  can  be  stopped. 

42.  Cases. — For  low  heads  turbines  may  be  used  in  open 
flumes  without  cases.  Fig.  4,  page  3,  Fig.  8,  page  10,  and 
Fig.  35,  page  41,  are  of  this  character.  Fig.  52  shows  such  a 
type  consisting  of  four  wheels  on  a  horizontal  shaft. 

Cases  may  also  be  used  for  very  low  heads  and  are  always 
used  for  high  heads.  The  cheapest  cases  are  the  cylinder  cases 
(Fig.  10,  page  12),  and  the  globe  cases  (Fig.  53).  These  cases 
are  undesirable  because  they  permit  of  considerable  eddy  loss 
as  the  water  flows  into  them  and  around  in  them  to  the  guides. 


THE  REACTION  \TURBINE 


60 


HYDRAULIC  TURBINES 


The  cone  case  shown  in  Fig.  54  is  a  very  desirable  type.  It  can 
be  seen  that  the  water  suffers  no  abrupt  changes  in  velocity  as  it 
flows  from  the  penstock  to  the  guides,  but  instead  is  uniformly 
accelerated. 

The  spiral  case,  illustrated  by  Fig.  55,  is  considered  the  best 
tvpe.  The  area  of  the  waterway  decreases  as  the  case  encircles 
the  guides,  because  only  a  limited  portion  of  the  water  flows  clear 


FIG.  53. — Turbine  in  globe  case.     (Made  by  James  Leffel  and  Co.) 

around  to  enter  the  further  part  of  the  circumference.  Thus  the 
average  velocity  throughout  the  case  is  kept  the  same.  The 
ca.se  is  also  designed  to  accelerate  the  water  somewhat  as  it 
leaves  the  penstock  and  flows  to  the  guides. 

Globe  and  spiral  cases  for  low  heads  are  made  of  cast  iron.  For 
higher  heads  they  are  made  of  cast  steel  as  in  Fig.  56.  Cylinder 
cases  (Fig.  10,  page  12),  are  usually^made  of  riveted  sheet  steel. 
Some  very  nice  spiral  cases  are  now  made  by  several  firms  of 


THE  REACTION  TURBINE 


61 


riveted  steel,  as  may  be  seen  in  Fig.  57.  Recent  practice  with 
large  vertical  shaft  units  is  to  form  the  case  of  reinforced  concrete 
as  in  Fig.  58.  The  weight  of  the  turbine  and  generator  is  carried 
by  the  "pit  liner/'  which  is  set  into  the  concrete,  and  this  in  turn 


FIG.  54. — Cone  case  turbine. 


rests  upon  the  " speed  ring."  The  latter  consists  of  an  upper  and 
a  lower  flange,  as  shown,  which  are  joined  together  by  vanes  so 
shaped  as  to  conform  to  the  free  stream  lines  of  the  water  flowing 
from  the  case  into  the  guide  vanes.  Thus  they  offer  less  re- 


62 


HYDRAULIC  TURBINES 


sistance  to  the  flow  of  the  water  than  the  round  columns  that 
were  once  employed.     The  speed  ring  vanes  are  also  shown  in 


[  FIG.  55. — Spiral  case  turbine 


FIG.  56. — Cast  steel  spiral  casings  at  Niagara  Falls.     14,000  h.p.  at  ISO  ft.  head. 
(Made  by  Wellman-Seaver-M organ  Co.) 

Figs.  59  and  60.     In  some  instances  the  case  is  of  sheet  metal 
surrounded  by  concrete  as  in  Fig.  60. 


THE  REACTION  TURBINE 


63 


43.  Draft  Tube  Construction. — It  must  be  borne  in  mind  that 
the  draft  tube  fulfills  two  distinct  functions.     First,  it  connects 


(Courtesy  of  S.  Morgan  Smith  Co.) 
FIG.  57.— Spiral  case  of  riveted  steel  plates. 


FIG.   58. — Typical  large  vertical  shaft  unit. 

the  turbine  runner  with  the  tail  water  and,  since  it  is  air  tight, 
it  prevents  any  loss  of  head  due  to  the  elevation  of  the  wheel. 


64 


HYDRAULIC  TURBINES 


Second,  it  may  be  made  to  reduce  the  outflow  loss  and  thus  to 
improve  the  efficiency  of  the  plant. 

For  the  first  purpose  alone  the  tube  might  be  made  of  a  uni- 
form cross-section,  but  in  practice  it  is  always  made  diverging 
so  as  to  accomplish  the  second  object  as  well.  In  fact,  even  if 
the  runner  should  be  set  below  the  tail-water  level,  a  draft  tube 
would  be  of  value  for  the  second  purpose.  This  was  proven 
many  years  ago  when  Francis  tested  an  outward-flow  turbine 
with  a  "diffuser"  surrounding  the  runner  and  found  that  the 


(Courtesy  of  Allis-Chalmers  Mfg.  Co.) 
FIG.  59. — Speed  ring. 

latter  improved  the  efficiency  by  3  per  cent.  As  has  been  pointed 
out_in  Art.  37,  the  higher  the  capacity  of  a  runner  of  given 
diameter  the  greater  the  velocity  of  the  water  must  be  at  the 
point  of  outflow  from  it  into  the  draft  tube.  This  velocity 
represents  kinetic  energy  which  would  otherwise  be  carried  away 
and  many  modern  wheels  of  an  extreme  high-capacity  type 
would  not  have  favorable  efficiencies  at  all  if  it  were  not  for  the 
use  of  a  suitable  draft  tube. 


THE  REACTION  TURBINE 


65 


The  reason  for  this  gain  in  efficiency  may  be  seen  in  either  of 
the  following  ways.  First,  the  total  power  available  is  that  due 
to  the  fall  from  head-water  level  to  tail-water  level.  The  power 
of  the  turbine  is  less  than  this  by  an  amount  equal  to  that  lost 
in  the  intake,  penstock,  and  draft  tube.  Anything  which  re- 
duces the  loss  outside  the  turbine  adds  just  that  much  more  to 


(Courtesy  of  Allis-Chalmers  Mfg.  Co.) 

FIG.  60.— Turbine  for  Niagara  Falls  Power  Co.,  37,500  h.p.,  214  ft.  head,  150 

r.p.m. 

the  power  which  the  water  can  give  up  within  the  turbine.  The 
velocity  head  with  which  the  water  leaves  the  lower  end  of  the 
draft  tube  represents  kinetic  energy  which  is  lost,  and  the  less 
this  discharge  loss  becomes,  the  better  the  efficiency  of  the  tur- 
bine. From  another  standpoint  the  pressure  at  the  upper  end 
of  the  draft  tube  depends  not  only  upon  the  elevation  of  this 

6 


66 


HYDRAULIC  TURBINES 


point  above  the  tail-water  level  but  also  upon  the  velocity  of  the 
water  at  that  section  and  the  losses  within  the  tube,  including 
the  discharge  loss  at  its  mouth.  The  less  this  loss,  the  lower 
the  pressure  at  the  upper  end.  And  the  less  the  pressure  at  the 
point  of  outflow  from  the  turbine  runner,  the  better  will  be  its 
performance. 

Draft  tubes  are  usually  made  of  riveted  steel  plates  as  in  Fig. 
10,  page  12,  or  are  moulded  in  concrete  as  in  Fig.  8,  page  10. 


FIG.  61. — Draft  tube  with  quarter  turn. 

The  tube  should  preferably  be  straight  but  where  the  setting 
does  not  permit  of  enough  room  for  this  without  excessive  cost 
of  excavation  the  tube  is  often  turned  so  as  to  discharge  hori- 
zontally as  in  Fig.  61.  If  the  tube  is  large  in  diameter  it  may  be 
necessary  to  make  the  horizontal  portion  of  some  other  section 
than  circular  as  in  Fig.  62,  in  order  that  the  vertical  dimension 
may  not  be  too  great.  A  good  form  of  section  to  use  is  oval. 
The  draft  tube  is  commonly  made  as  a  frustum  of  a  cone  with  a 


• 
THE  REACTION  TURBINF  67 

vertex  angle  of  8°.  If  the  section  becomes  some  other  shape, 
the  tube  is  so  made  that  the  area  increases  at  a  similar  rate  to 
what  it  would  if  it  were  circular.  The  conical  form  has  been 
largely  employed  chiefly  because  of  ease  of  manufacture,  but 
when  draft  tubes  are  moulded  in  concrete  other  forms  may  be 
used.  A  form  that  is  theoretically  good  is  " trumpet  shaped," 
somewhat  as  in  Fig.  60,  so  that  the  velocity  of  the  water  may  be 
made  to  decrease  uniformly  along  the  length  of  the  tube.  In 
any  event  the  draft  tube  should  be  so  made  as  to  secure  a  gradual 
reduction  of  velocity  from  the  runner  to  the  mouth. 


(Courtesy  of  Wellman-Seaver-M organ  Co.') 
FIG.  62.— Mouth  of  draft  tube  at  Cedar  Rapids. 

The  most  recent  innovation  in  draft  tube  construction  is  shown 
in  Figs.  60  and  63.  At  the  lower  end  of  a  comparatively  short 
draft  tube  is  a  conoidal  portion  through  which  the  water  passes 
just  before  impinging  on  a  circular  plate  which  is  concentric 
with  the  tube.  The  water  turns  and  flows  out  along  this  plate 
around  its  entire  circumference  through  an  annular  opening 
into  a  collecting  chamber  and  from  thence  through  a  horizontal 
diverging  tube  to  the  tail  race.  As  the  water  flows  through 
the  conoidal  portion  of  the  tube  and  impinges  on  the  plate,  its 
velocity  is_greatly  reduced.  This  portion  is  called  by  the  in- 


68 


HYDRAULIC  TURBINES 


ventor,  W.  M.  White,  a  "hydraucone."  As  the  water  turns 
and  flows  through  the  annular  opening,  its  velocity  is  increased 
and  is  then  decreased  again  as  it  enters  the  collecting  chamber. 
The  velocity  is  still  further  decreased  as  the  water  flows  to  the  tail 
race  through  the  horizontal  tube.  A  design  by  Lewis  F.  Moody 
differs  from  the  above  in  that  the  collecting  chamber  is  a  spiral, 
somewhat  like  the  spiral  case,  so  proportioned  that  the  water  is 
continuously  decelerated  throughout  the  flow. 

Bearing  in  mind  that  one  function  of  the  draft  tube  is  to 
efficiently  convert  velocity  head  into  pressure  head,  we  see  the 


FIG.  63. — Draft  tube  with  hydraucone. 

limitations  of  the  ordinary  construction.  In  order  to  secure  the 
diffusion  desired,  the  length  of  the  tube  may  be  such  that  the 
expense  of  excavation  is  prohibitive  and  hence  the  tube  is  turned 
from  vertical  to  horizontal  with  a  bend  of  short  radius.  But  such 
a  bend  inevitably  induces  eddy  losses  which  interfere  with  the 
efficient  performance  of  the  tube.  Furthermore  velocity  head 
cannot  usually  be  converted  into  pressure  head  without  a  great 
deal  of  loss  unless  the  flow  of  the  water  be  smooth.  Since  the 
discharge  from  a  turbine  runner  is  usually  quite  turbulent,  this 
alone  would  limit  the  value  of  a  draft  tube,  even  if  it  were 
straight.  If  the  device  just  described  is  properly  proportioned, 
then,  as  the  result  of  hydrodynamic  laws  for  which  we  have  not 


THE  REACTION  TURBINE 


69 


room  here,1  the  flow  may  be  turned  from  vertical  to  horizontal 
in  a  very  small  space  with  less  loss  of  energy  than  otherwise. 
Another  peculiarity  of  the  hydraucone  is  that  it  can  convert 
the  velocity  head  of  turbulent  water  into  pressure  head  efficiently. 
Then  when  this  water  is  accelerated,  as  it  leaves  through  the 
annular  opening,  it  flows  away  with  smooth  stream  lines  and  is 
in  proper  shape  for  the  ultimate  conversion  in  the  diverging 
horizontal  tube.  One  of  the  things  which  limits  the  high-speed 
high-capacity  type  of  runner  is  the  inability  of  the  draft  tube  to 
recover  the  kinetic  energy  of  the  water  leaving  the  runner, 
especially  in  view  of  the  fact  that  with  this  type  the  water  leaves 
with  some  considerable  "whirl."  This  new  development  may 


r* Case **-*G-uldesH<5 H 

j  Runnei*' 

FIG.  64. — Velocity  and  energy  transformations  in  turbine. 

make  it  possible  to  extend  the  present  limits  of  turbine  runner 
design. 

44.  Velocities. — The  velocities  at  different  points  are  indicated 
by  Fig.  64. 2  The  velocity  of  flow  in  the  penstock  is  determined 
by  the  consideration  of  the  cost  and  other  conditions  in  eace 
case.  The  mean  velocity  of  flow  allowable  in  the  turbine  cash 
is  as  follows: 

If  the  case  is  cylindrical  the  velocity  should  be  as  low  as  0.08 
to  O.l2\/2gh  where  h  is  the  effective  head.  If  a  spiral  case  is 
used  the  velocity  may  be  from  0.15  to  0.24\/2gh.  For  heads  of 
several  hundred  feet  the  value  of  0.15  is  used  to  reduce  wear  on 
the  case,  0.20  is  used  for  moderate  heads,  and  0.24  is  used  for 
low  heads. 

The  velocity  at  entrance  to  the  turbine  runner,  Vi  =  0.6  to 

\The  Journal  of  the  Association  of  Engineering  Societies,  vol.  27,  p.  39. 

2_Mead's  "Water  Power  Engineering." 


70  HYDRAULIC  TURBINES 

Q.8\/2gh.  The  velocity  at  the  point  of  discharge,  V%  =  from 
0. 10  to  QAQ\/2gh.  These  values  depend  entirely  upon  the  design 
of  the  turbine  and  are  not  arbitrarily  assigned. 

The  velocity  at  entrance  to  the  upper  end  of  the  draft  tube 
should  equal  the  velocity  with  which  the  water  leaves  the  turbine, 
otherwise  a  sudden  change  in  velocity  will  take  place.  Velocity 
of  discharge  from  the  lower  end  of  the  draft  tube  may  be  about 
0.10  to  Q.15\/2gh.  The  value  of  the  latter  is  determined  by  the 
value  of  the  velocity  at  the  upper  end  and  by  the  length  and 
the  amount  of  flare  to  be  given  the  tube. 

45.  Conditions  of  Use. — The  reaction  turbine  is  best  adapted 
for  a  low  head  or  a  relatively  large  quantity  of  water.  As  was 
stated  in  Art  32,  the  choice  of  a  turbine  is  a  function  of  capacity 
as  well  as  head.  For  a  given  head  the  larger  the  horse-power 
the  more  reason  there  will  be  for  using  a  reaction  turbine. 

The  use  of  a  reaction  turbine  under  high  heads  is  accompanied 
by  certain  difficulties.  It  is  necessary  to  build  a  case  which  is 
strong  enough  to  stand  the  pressure;  also  the  case,  guides,  and 
runner  may  be  worn  out  in  a  short  time  by  the  water  moving  at 
high  velocities.  This  depends  very  much  upon  the  quality  of  the 
water.  Thus  a  case  is  on  record  where  a  wheel  has  been  operating 
for  six  years  under  a  head  of  260  ft.  with  clear  water  and  the  tur- 
bine is  still  in  excellent  condition.  Another  turbine  made  by  the 
same  company  and  according  to  the  same  design  was  operated 
under  a  head  of  160  ft.  with  dirty  water.  In  four  years  it  was 
completely  worn  out  and  was  replaced  with  an  impulse  wheel. 
The  tangential  water  wheel  has  the  advantage  that  the  relative 
velocity  of  flow  over  its  buckets  is  less  for  the  same  head  and  thus 
the  wear  is  less.  Also  repairs  can  be  more  readily  made. 

The  runners  of  reaction  turbines  and  the  buckets  of  impulse 
wheels  will  not  last  long  if  their  design  is  imperfect.  This  is  due 
to  the  fact  that  wherever  there  is  an  eddy  or  wherever  there  is  a 
point  of  extremely  low  pressure,  the  air  that  is  in  solution  in  the 
water  will  always  tend  to  be  liberated  at  that  point.  And  as 
water  tends  to  absorb  more  oxygen  in  proportion  to  nitrogen 
than  is  in  the  air,  the  result  is  that  the  liberated  mixture  is  rich 
in  oxygen  and  hence  readily  attacks  and  pits  the  metal.  Fig. 
6,  page  5,  shows  a  turbine  runner  that  has  had  holes  eaten  in 
it  because  of  this  reason.  Thus  a  defective  design  not  only 
produces  a  runner  of  lower  efficiency  because  of  the  eddy  losses 
within  the  water,  but  such  eddies  shorten  the  life  of  the  wheel. 


THE  REACTION  TURBINE  71 

Also  great  care  should  be  used  in  designing  so  that  the  velocity 
along  any  stream  line  does  not  cause  the  pressure  to  approach  the 
vapor  pressure  of  the  water  too  closely,  otherwise  the  same  action 
will  take  place.  With  the  reaction  turbine  it  is  possible  to  de- 
sign a  runner  free  from  eddies  for  one  gate  opening  only.  The 
operation  of  reaction  turbines  at  part  gate  for  long  periods  of 
time  must  inevitably  shorten  the  life  of  the  runner. 


FIG.  65. —22,. 500  h.p.  turbine  for  Pacific  Coast  Power  Co. 
(Made  by  Allis-Chatmers  Mfg.  Co.) 

Reaction  turbines  are  used  under  very  low  heads  in  som? 
instances.  The  lowest  head  on  record  is  16  in.  but  several  feet 
is  the  usual  minimum.  The  highest  head  yet  employed  for  a 
reaction  turbine  is  800  ft.  The  latter  is  used  for  two  22,500 
h.p.  units  built  by  the  Pelton  Water  Wheel  Co. 

The  most  powerful  turbine  in  the  world  will  develop  52,500 


72  HYDRAULIC  TURBINES 

h.p.  Two  such  units,  built  by  the  Allis-Chalmers  Mfg.  Co., 
will  be  installed  at  the  Chippewa  Development  of  the  Hydro- 
Electric  Power  Commission  of  Ontario.  The  head  is  320  ft. 

In  Fig.  60  is  shown  ji  turbine  of  37,500  h.p.,  which  is  similar 
in  type  to  the  above.  This  wheel  runs  under  a  head  of  214  ft. 
and  is  for  the  Niagara  Falls  Power  Co. 

There  are  at  present  quite  a  number  of  turbines  in  operation 
whose  power  ranges  from  20,000  to  30,000  h.p. 

The  power  of  a  turbine  depends  not  only  upon  its  size  but  also 
upon  the  head  under  which  it  operates.  The  turbines  above  are 
the  most  powerful,  but  they  are  not  the  largest  in  point  of  size. 
The  largest  turbines  so  far  are  the  10,800  h.p.  turbines  of  the 
Cedars  Rapids  (Canada)  Mfg.  and  Power  Co.,  which  run  at 
56.6  r.p.m.  under  a  head  of  30  ft.  The  rated  diameter  is  143  in., 
but  the  maximum  diameter' (see  Fig.  40)  is  17  ft.  8  in.  The 
runner  weighs  160,000  lb.,  the  revolving  part  of  the  generator 
and  the  shaft,  390,000  lb.,  while  the  suspension  bearing  weighs 
300,000  lb.  The  total  weight  of  the  entire  unit  is  1,615,000  lb. 

The  largest  runners  in  this  country  are  those  of  the  Mississippi 
River  Power  Co.  at  Keokuk,  la.  They  develop  10,000  h.p. 
at  57.7  r.p.m.  under  a  head  of  32  ft.  They  are  slightly  smaller 
than  those  at  Cedars  Rapids.  (See  Figs.  42  and  47.)  The  I.  P. 
Morris  Co.  built  eight  of  these  wheels  and  the  Wellman-Seaver- 
Morgan  Co.  seven.  These  two  concerns  likewise  built  the 
Cedars  Rapids  turbines. 

46.  Efficiency. — The  efficiency  obtained  from  the  average 
reaction  turbine  may  be  from  80  to  85  per  cent.  Under  favorable 
conditions  with  large  capacities  higher  efficiencies  up  to  about  90 
per  cent,  or  more  may  be  realized.  For  small  powers  or  un- 
favorable conditions  75  per  cent,  is  all  that  should  be  expected. 

47.  QUESTIONS 

1.  What  was  the  origin  of  the  Fourneyron  turbine  ?     What  is  the  Jonval 
turbine?     What  was  the  origin  of  the  Francis  turbine? 

2.  What  is  the  Swain  turbine?     What  is  the  McCormick  turbine?     Why 
were  they  developed?     What  is  the  modern  Francis  turbine?     Why  is  this 
name  attached  to  all  inward  flow  turbines  at  present? 

3.  Sketch  the  profiles  of  different  types  of  modern  turbine  runners  and 
explain  why  they  are  so  built. 

4.  Why  has  the  inward  flow  turbine  superseded  the  outward  flow  turbine  ? 

5.  How  does  the  angle  a\  vary  with  different  types  of  runner  and  why? 
How  does  the  factor  <f>«  vary  and  why? 


THE  REACTION  TURBINE  73 

6.  For  a  given  head  and  diameter  of  runner  explain  how  the  power  varies 
with  different  types.     For  a  given  head  and  power  explain  how  the  rotative 
speed  varies. 

7.  Draw  typical  vector  diagrams  for  the  velocities  at  entrance  to  the  dif- 
ferent types  of  turbine  runners.     Show  how  the  vane  angle  varies. 

8.  How  are  turbine  runners  constructed?     What  materials  are  used? 
What  classes  of  runners  are  there  ? 

9.  What  are  the  different  kinds  of  gates  used  for  governing  reaction  tur- 
bines, and  what  are  their  relative  merits  ? 

10.  What  means  are  provided  to  save  the  penstock 'from  water  hammer, 
when  the  gates  of  a  reaction  turbine  are  quickly  closed  ?     How  are  the  gates 
of  a  turbine  operated? 

11.  What  kinds  of  bearings  are  used  for  horizontal  shaft  turbines?     For 
vertical  shaft  turbines?     What  means  may  be  provided  to  take  care  of  end 

hrust  in  either  type? 

12.  What  types  of  cases  are  used  for  turbines?     What  are  the  cheapest 
forms  and  what  are  the  best?     What  are  speed  rings? 

13.  What  is  the  purpose  of  a  draft  tube  and  how  are  they  constructed  ? 

14.  What  different  factors  may  cause  a  turbine  runner  to  wear  out? 
Under  what  range  of  heads  are  reaction  turbines  now  used? 

16.  What  horsepower  is  developed  by  the  most  powerful  turbine? 
What  is  the  largest  in  point  of  size?  Why  is  not  the  largest  one  also  the 
most  powerful?  What  efficiencies  should  be  expected  from  reaction 
turbines? 


CHAPTER  VI 
TURBINE  GOVERNORS 

48.  General  Principles. — All  governors  depend  primarily  upon 
the  action  of  rotating  weights.  Thus  the  governor  head  in  Fig.  66 
is  rotated  by  some  form  of  drive  so  that  its  speed  is  directly 
proportional  to  that  of  the  machine  which  it  regulates.  The 
higher  the  speed  of  rotation,  the  farther  the  balls  stand  from 
the  axis,  and  the  higher  will  the  collar  be  raised  on  the  vertical 
spindle.  The  collar  in  turn  transmits  motion  to  some  element 
of  the  mechanism  which  effects  the  speed  regulation. 

Let  W  be  the  weight  of  each  ball,  2KW  that  of  the  center 
weight,  h  the  height  of  the  "cone"  in  inches,  x  the  ratio  of  the 


w 


FIG.  66. — Governor  head. 

velocity  of  the  collar  to  the  vertical  velocity  of  the  balls,  and  N 
the  revolutions  per  minute  of  the  governor  head.  Also  let  the 
force  which  opposes  the  motion  of  the  collar,  due  to  the  friction 
of  the  moving  parts  of  the  governor  mechanism  actuated  by  it, 
be  denoted  by  2fW.  Then  the  following  equation  may  be  found 
to  hold : 

N2h  =  35,200  [1  +  x(K  ±  /)]. 

Considering  the  right  hand  member  of  the  above  as  constant  for 
the  moment,  it  may  be  seen  that  for  every  value  of  h  there  must 
be  a  definite  value  of  N.  For  different  loads  on  the  machine  it 

74 


TURBINE  GOVERNORS 


75 


is  necessary  that  the  gates  and  gate  mechanism  occupy  different 
positions  and,  if  this  requires  different  positions  of  the  collar  of 
the  governor  head,  it  may  be  seen  that  the  speed  must  decrease 
as  the  load  increases.1 

If  the  change  of  speed  from  no  load  to  full  load  be  denoted  by 
AJV  and  N  be  interpreted  as  the  average  speed,  the  coefficient  of 
speed  regulation  is  AN/N.  This  coefficient  may  be  reduced  to  a 
very  small  value  by  careful  design  of  the  governor.  The  essen- 
tials of  a  good  governor  are: 

1.  Close  regulation  or  a  small  value  of  &N/N. 

2.  Quickness  of  regulation. 

3.  Stability  or  lack  of  hunting. 

4.  Power    to    move    parts    or    to    resist    disturbing    forces. 
To  some  extent  certain  of  these  requirements  conflict  with  others 


N 
FIG.  67. 

so  that  the  final  design  is  something  of  a  compromise.  Close 
regulation  may  be  obtained  by  so  proportioning  the  arms  that 
#  is  a  variable  in  such  a  way  as  to  .permit  h  to  change  but  little 
for  the  different  collar  positions.  Stability  and  power  may  be 
secured  by  making  the  center  weight,  2KW,  sufficiently  heavy. 
This  weight  is  often  replaced  by  a  spring,  which  exerts  an 
equivalent  force.  The  importai  ce  of  a  large  value  of  K  is  seen 
when  we  consider  its  relation  to  the  friction.  The  latter  changes 
sign  according  to  the  direction  of  motion,  and  may  also  change 

1  A  constant  speed  or  asynchronous  governor  could  be  constructed  by  so 
arranging  it  that  h  remained  constant  as  the  balls  changed  their  position, 
but  such  a  governor  would  lack  stability  as  it  might  be  in  equilibrium  with 
the  collar  in  any  position  for  a  given  speed.  Then  for  a  slight  change  in 
speed  the  governor  would  move  over  to  its  extreme  position  which  would 
be  limited  by  a  stop.  Such  governors  have  been  built,  however,  and  are 
practicable  if  a  strong  dash  pot  is  used  to  prevent  their  "hunting." 


76  HYDRAULI'C  TURBINES 

in  value  from  time  to  time,  so  that  the  larger  the  value  of  K 
the  smaller  will  be  the  effect  of  friction,  arid  the  closer  together 
will  the  two  curves  of  Fig.  67  be.  We  can  also  see  the  great 
necessity  for  keeping  /  as  small  as  possible,  and  this  requirement 
leads  us  to  the  use  of  the  relay  governor. 

The  operation  of  the  nozzles  of  impulse  wheels  or  of  the  gates 
of  reaction  turbines  requires  a  considerable  force  to  be  exerted. 
The  governor  head  could  not  do  this  directly  without  being  of 
prohibitive  size  and  hence  it  does  nothing  more  than  set  some 
relay  device  into  action,  the  latter  furnishing  the  power  to 
operate  the  regulating  mechanism. 

49.  Types  of  Governors. — There  are  two  fundamental  types  of 
water  wheel  governors : 

(a)  Mechanical  governors. 

(6)  Hydraulic  governors. 

With  the  first  type  the  governor  head  causes  some  form  of 
clutch  to  be  engaged  so  that  the  gates  are  operated  by  the  power 
of  the  turbine  itself.  This  is  the  least  expensive  but  has  the 
disadvantage  that  the  operation  of  the  gates  adds  just  that  much 
more  to  a  demanded  load.  The  second  type  of  governor  costs 
more  but  is  always  used  in  the  best  plants.  The  governor  head 
in  this  case  merely  operates  a  pilot  valve  which  admits  a  liquid 
under  pressure  to  one  side  or  the  other  of  a  piston  in  a  cylinder. 
This  piston  and  cylinder  is  known  as  the  servo-motor  and 
operates  the  gates. 

The  liquid  used  to  operate  the  servo-motor  is  stored  under  air 
pressure  in  a  tank  into  which  it  is  pumped.  The  power  for 
operating  the  gates  also  comes  from  the  turbine  in  this  instance 
but  it  is  spread  over  the  entire  period  of  operation  instead  of 
being  concentrated  just  when  the  load  is  changing.  Oil  is 
commonly  used  as  the  working  fluid  and  is  very  satisfactory 
except  for  its  cost.  Some  effort,  which  is  meeting  with  success, 
is  being  made  to  produce  emulsions  consisting  principally  of 
water  but  which  will  be  similar  to  oil  in  its  action.  If  water 
alone  is  used,  it  should  be  carefully  filtered  and  circulated  over 
and  over  again.  Occasionally  water  has  been  used  under  pen- 
stock pressure  and,  of  course  direct  from  the  penstock,  but  the 
grit  and  sediment  in  it  is  very  bad  for  the  operating  parts  of 
the  governor. 

50.  The  Compensated  Governor. — It  has  already  been  pointed 
out  in  Art.  26  that  the  inertia  of  the  water  in  the  penstock  and 


TURBINE  GOVERNORS 


77 


draft  tube  makes  close  speed  regulation  difficult,  since  often  the 
immediate  result  of  a  change  of  gate  position  is  directly  opposite 
to  that  desired.  With  the  simple  type  of  governor  the  latter 
would  continue  to  operate  in  the  same  direction  under  these 
circumstances  and  would  thus  move  far  beyond  the  proper 
point.  Finally  when  the  hydraulic  conditions  would  readjust 
themselves  the  governor  would  then  be  compelled  to  move  back, 
but  would  this  time  pass  to  the  other  side  of  the  proper  place. 
Thus  the  governor  would  continually  "hunt"  and  maintain  a 
constant  oscillation  of  flow  in  the  pipe  line. 


To  Kegulate     Speed 

variation  from  No  Load 

to  Full  Load 

FIG.  68. — Compensated  floating  lever  governor. 

To  prevent  such  action  a  waterwheel  governor  is  usually 
"compensated"  so  that  it  will  slowly  approach  its  proper  place 
and  practically  remain  there.  Such  a  governor  is  also  said  to  be 
"dead  beat."  In  Fig.  68  may  be  seen  the  essential  features  of 
the  floating-lever  compensated  governor.  If,  for  example,  the 
wheel  speed  increases,  the  balls  of  the  governor  raise  the  collar, 
C,  to  which  is  attached  the  floating  lever.  The  latter  for  the 
moment  pivots  about  A  and  through  the  link  at  B  raises  the  relay 
valve,  D.  This  action  admits  oil  (or  other  fluid)  to  the  left- 
hand  side  of  the  servo-motor  piston  and  exhausts  it  from  the 
right-hand  side,  thus  compelling  the  piston  to  move  to  the  right 
and  decrease  the  turbine  gate  opening.  But  at  the  same  time 


78 


HYDRAULIC  TURBINES 


the  bell-crank  EFG,  which  is  attached  to  the  gate  connecting 
rod  at  E,  is  rotated  about  F  so  the  arm  G  is  lowered.  This  pulls 
down  the  pivot  A,  which  causes  B  to  be  lowered,  thus  closing  the 
relay  valve  ports  and  stopping  the  motion.  Thus  the  governor 
is  prevented  from  over-travelling.  Of  course,  if  the  gates  have 
not  been  moved  far  enough,  this  action  can  be  repeated. 

The  dash  pot,  H ,  will  not  cause  the  pivot  A  to  be  moved  unless 
the  governor  acts  quickly.  If  the  governor  changes  slowly, 
there  is  little  need  for  the  compensating  action  and  hence  the 
dash  pot  does  not  then  transmit  the  motion.  But  there  is  a 


FIG.  69. 


(Courtesy  of  Allis-Chalmers  Mfg.  Co.) 
Hydraulic  turbine  governor. 


second  rod  from  G  which  is  connected  with  the  other  vei  tical  rod 
by  springs  at  M.  This  will  serve  to  stop  the  motion  in  such  a 
case  though  it  does  not  move  A  as  much,  since  it  has  a  shorter 
radius  arm. 

For  a  given  speed  of  the  governor  head,  and  hence  for  a  given 
position  of  the  collar  C,  moving  A  will  tend  to  shift  the  relay 
valve  and  hence  change  the  position  of  the  turbine  gates.  But 
if  the  turbine  gates  are  changed,  without  any  corresponding 
change,  in  load,  the  turbine  speed  will  vary.  The  length  of  the 
rod  from  G  to  M  is  adjustable  by  turning  a  wheel  L  into  which 
the  two  ends  of  the  rods  are  fitted  with  right  and  left  handed 
threads.  Hence  the  speed  of  the  turbine  can  be  varied  within 


TURBINE  GOVERNORS  79 

certain  limits  by  L,  which  is  convenient  for  synchronizing,  for 
instance.  Generally  L  is  turned  by  a  very  small  electric  motor 
which  can  be  operated  from  the  switchboard. 

At  full-load  on  the  turbine  the  servo-motor  piston  is  at  the 
opposite  end  of  the  stroke  from  no-load  and  hence  the  pivot  A 
has  a  corresponding  vertical  travel.  The  amount  of  this  travel 
can  be  altered  by  changing  the  radius  of  the  connection  at  G. 
Considering  B  as  fixed  (as  it  must  be  if  the  relay  valve  is  closed  in 
both  cases)  it  is  evident  that  changing  the  amount  of  travel  of 
A  will  change  the  amount  of  travel  of  the  collar  C.  Remember- 
ing that  different  positions  of  collar  C  correspond  to  definite 
values  of  N,  it  is  clear  that  changing  the  amount  of  travel  of  the 
collar  C  from  no-load  to  full-load  will  vary  the  speed  regulation. 

Other  adjustments  that  can  be  made  to  secure  the  proper 
degree  of  sensitiveness  for  the  hydraulic  conditions  are  to  vary 
the  springs  at  M  and  to  change  the  speed  of  the  dash  pot. 

One  of  the  recent  changes  in  governor  construction  for  vertical 
type  turbines  is  to  mount  the  rotating  weight  on  the  turbine  shaft 
itself.  This  eliminates  any  lost  motion  between  the  turbine  and 
governor  head. 

51.  QUESTIONS 

1.  With  the  usual  type  of  governor,  why  must  the  speed  vary  to  a  slight 
extent  from  no-load  to  full-load?     Which  way  does  the  speed  change  as 
the  load  increases?     Why? 

2.  What  qualification    are  essential  in  a  good  governor  and  how  may 
they  be  obtained?     What  is  the  effect  of  friction  on  the  operation  of  the 
governor? 

3.  Why  is  the  speed  range  for  a  decreasing  load  different  from  that  for 
an  increasing  load  ?     What  is  the  purpose  of  the  center  weight  or  the  spring 
loading  in  governors? 

4.  What  is  a  relay  governor?     Why  is  it  necessary  for  water  wheels? 
How  is  it  operated? 

5.  What  are  the  relative  merits  of  different  types  of  relay  governors? 
What  are  the  relative  merits  of  the  fluids  used  in  hydraulic  governors? 

6.  What  is  the  compensated  governor?     Why  is  it  necessary?     Describe 
the  action  of  one? 

7.  Describe  the  adjustments  that  can  be  made  on  a  floating  lever  governor 


CHAPTER  VII 
GENERAL  THEORY 

52.  Equation  of  Continuity. — In  a  stream  with  steady  flow 
(conditions  at  any  point  remaining  constant  with  respect  to 
time)  the  equation  of  continuity  may  be  applied.     This  is  that 
the  rate  of  discharge  is  the  same  for  all  cross-sections  so  that 
q  =  AV  =  av  =  constant,  and  in  particular 

q  =  AiVi  =  aiVi  =  a&z  (1) 

53.  Relation  between  Absolute  and  Relative  Velocities. — The 

absolute  velocity  of  a  body  is  its  velocity  relative  to  the  earth. 
The  relative  velocity  of  a  body  is  its  velocity  relative  to  some 
other  body  which  may  itself  be  in  motion  relative  to  the  earth. 
The  absolute  velocity  of  the  first  body  is  the  vector  sum  of  its 
velocity  relative  to  the  second  body  and  the  velocity  of  the  second 
body.  The  relation  between  the  three  velocities  u,  v,  V  is  shown 


FIG.  70.  —  Relation  between  relative  and  absolute  velocities. 

by  the  vector  triangles  in  Fig.  70.     The  tangential  component  of 
Fis 

Vu  —  V  cos  A  =  u  +  v  cos  a  (2) 

54.  The  General  Equation  of  Energy.  —  Energy  may  be  trans- 
mitted across  a  section  of  a  flowing  stream  in  any  or  all  of  the 
three  forms  known  as  potential  energy,  kinetic  energy,  or  pressure 
energy.1  Since  head  is  the  amount  of  energy  per  unit  weight 
of  water,  the  total  head  at  any  section 


<»> 


M.  Hoskins,  "Hydraulics,"  Chapter  IV. 

80 


GENERAL  THEORY 


81 


There  can  be  no  flow  without  some  loss  of  energy  so  that  the  total 
head  must  decrease  in  the  direction  of  flow  by  the  amount  of  head 
lost  or 

F!  -  #2  =.  Head  lost  (4^ 

Suffixes  (1)  and  (2)  may  here  denote  any  two  points. 

In  flowing  through  the  runner  of  a  turbine  the  water  gives  up 
energy  to  the  vanes  in  the  form  of  mechanical  work  and  a  portion 
of  the  energy  is  lost  in  hydraulic  friction  and  is  dissipated  in  the 
form  of  heat.  Thus  the  head'lost  by  the  water  equals  In!'  +  hf. 
And  if  suffixes  (1)  ;  and  (2)  "are  restricted  to  the  points  of 
entrance  to  and  discharge  from  the  runner,  equation  (4)  may 
be  written 


(5) 


55.  Effective  Read  on  Wheel. — Obviously  the  turbine  should 
not  be  charged  up  with  head  which  is  lost  in  the  pipe  line,  so  the 


FIG.  71. — Effective  head  for  tangential  water  wheel. 

value  of  h  should  be  the  total  fall  available  minus  the  penstock 
losses.  Thus  if  Z  is  total  fall  available  from  head  water  to  tail 
water,  Hf  the  head  lost  in  the  penstock  or  other  places  outside 
the  water  wheel,  and  h  the  net  head  actually  supplied  the  turbine, 
we  have 

h  =  Z  -  H'  (6) 

The  head  supplied  to  the  impulse  wheel  in  Fig.  71  is  the  head 
measured  at  the  base  of  the  nozzle.  Thus  for  the  tangential 
water  wheel 


H, 


w 


29 


(7) 


The  reaction  turbine,  shown  in  Fig.  72,  is  able  to  use  the 
total  fall  to  the  tail-water  level  by  virtue  of  its  employment  of 
the  draft  tube.  Hence  the  total  head  supplied  to  the  wheel  at  C 


82 


HYDRAULIC  TURBINES 


is  measured  above  the  tail-water  level  as  a  datum  plane, 
for  the  reaction  turbine 


Thus 


(8) 


The  turbine  with  its  draft  tube,  which  in  a  sense  is  as  much 
an  appurtenance  of  the  runner  as  the  guide  vanes,  is  here  charged 
with  the  total  amount  of  the  energy  supplied  to  it.  The  kinetic 
energy  of  the  water  at  discharge  from  the  mouth  of  the  draft  tube 
E  is  a  loss  for  which  the  runner  and  draft  tube  may  be  said  to  be 
responsible  in  part,  though  some  loss  there  is  inevitable,  but  the 
trouble  is  that  the  setting  of  the  turbine,  over  which  the  turbine 
builder  has  little  control,  limits  the  design  of  the  draft  tube  and 


FIG.  72. — Effective  head  for  reaction  turbine. 

hence  the  manufacturer  may  not  be  able  to  reduce  this  discharge 
loss  to  a  desired  value.  Two  similar  runners  installed  under 
different  settings  might  yield  different  efficiencies  because  of  this. 
Consequently  turbine  builders  desire  some  method  which  will 
make  the  measured  efficiency  of  a  runner  independent  of  the 
conditions  of  the  setting  over  which  their  designers  have  no 
control.  This  second  method  is  to  charge  up  the  turbine  with  all 
losses  within  the  draft  tube  but  to  credit  it  with  the  velocity 
head  at  the  point  of  discharge.  Thus 


h  =  Hc  —  HE  —  zc 


PC 
W 


v  c 

2S 


(9) 


It  is  believed  that  equation  (8)  is  rational  and  scientifically 
correct,  but  that  equation  (9)  may  be  commercially  more  de- 
sirable.1 In  general  the  actual  numerical  difference  between 
the  values  of  h  computed  by  these  two  methods  will  be  small. 

1  For  discussion  on  this  point  see  "Investigation  of  the  Performance  of  a 


GENERAL  THEORY  83 

56.  Power  and  Efficiency. — Since  head  is  the  amount  of  energy 
per  unit  weight  of  water  it  follows  that  by  multiplying  by  the 
total  weight  of  water  per  unit  time  we  have  energy  per  unit  time 
and  this  is  power.  Thus 

Power  =  WH  =  pounds  per  second  X  feet  (10) 

In  this  expression  H  may  be  interpreted  as  in  (3)  or  it  may  be 
replaced  by  h"  or  any  other  head  according  to  what  is  wanted. 

But  also  power  equals  force  applied  times  the  velocity  of  the 
point  of  application.  Thus 

Power  =  Fu  =  pounds  X  feet  per  second  (11) 

where  F  represents  the  total  force  applied. 
Torque,  T,  equals  F  X  r  and  angular  velocity  o>  = 
Since  then  Fu  =  Tu  it  is  evident  that  , 

Power  =  Tco  =  foot  pounds  per  second   I  (12) 

Any  of  these  three  expressions  for  power  may  be  used  according 
to  circumstances.  While  (11)  is  the  most  obvious  to  many,  it 
will  be  found  that  in  hydraulics  (10)  is  usually  more  convenient. 

(The  following  simplifications  for  horsepower  of  a  turbine  are 
convenient.  Using  the  h  of  Art.  55, 

B.h.p.  =  62.5  qhe/550  =  qhe/8.8. 

For  estimations,  the  value  of  the  efficiency  may  be  assumed  as 
0.80  in  which  case  our  expression  becomes  h.p.  =  qh/ll. 

The  word  " efficiency"  is  always  understood  to  mean  total 
efficiency.  It  is  the  ratio  of  the  developed  or  brake  power  to 
the  power  delivered  in  the  water  to  the  turbine  based  on  the 
head  h  of  Art.  55. 

Reaction  Turbine,"  by  R.  L.  Daugherty,  Trans.  Am.  Soc.  of  C.  E.,  vol.  78, 
p.  1270  (1915). 

It  may  be  noted  that  it  might  be  desirable  under  some  circumstances  to 
eliminate  the  draft  tube  losses  altogether  and  compute  the  efficiency  of  the 
runner  alone.  This  would  necessitate  the  measurement  of  the  head  at  D 
in  order  to  take  the  difference  between  it  and  the  head  at  C.  The  practical 
difficulty  here  is  that,  due  to  the  turbulent  and  often  rotary  motion  of  the 
water  at  this  point,  it  is  impossible  to  measure  the  pressure  with  any  degree 
of  accuracy.  Likewise  the  velocity  head  cannot  be  computed,  since  the  actual 
velocity  under  the  conditions  of  flow  will  not  be  equal  to  the  rate  of  discharge 
divided  by  the  cross-section  area.  This  same  consideration  holds  in  regard 
to  the  computation  of  the  velocity  head  at  E. 


84 


HYDRAULIC  TURBINES 


Mechanical  efficiency  is  the  ratio  of  the  power  delivered  by 
the  machine  to  that  delivered  to  its  shaft  by  the  runner.  The 
difference  between  these  two  powers  is  that  due  to  mechanical 
losses. 

Hydraulic  efficiency  is  the  ratio  of  the  power  actually  de- 
livered to  the  shaft  to  that  supplied  in  the  water  to  the  runner. 
The  difference  between  these  two  is  due  to  hydraulic  losses. 

Volumetric  efficiency  is  the  ratio  of  the  water  actually  passing 
through  the  runner  to  that  supplied.  The  difference  between 
these  two  quantities  is  the  leakage  through  the  clearance  spaces. 

The  total  efficiency  is  the  product  of  these  three.     Thus 

e  =  em  X  eh  X  e0. 


57.  Force  Exerted. — Whenever  the  velocity  of  a  stream  of 
water  is  changed  either  in  direction  or  in  magnitude,  a  force  is 
required.  By  the  law  of  action  and  reaction  an  equal  and 
opposite  force  is  exerted  by  the  water  upon  the  body  producing 
this  change.  This  is  called  a  dynamic  force. 

Let  R  be  the  resultant  force  exerted  by  any  body  upon  the 
water  and  Rx  and  Ry  be  its  components  parallel  to  x  and  y  axes. 
Also  let  us  here  consider  a  as  the  angle  made  by  V  with  the  x 
axis.  The  force  exerted  by  the  water  upon  the  body  will  be 
denoted  by  F.  Its  value  may  be  found  in  either  of  the  two  ways 
shown  below.  The  first  depends  upon  the  principle  that  the 
resultant  of  all  the  external  forces  acting  on  a  body  is  equal  to 

dV 
the  mass  times  the  acceleration  or  R  =  m  ^-     The  second  is 

based  upon  the  principle  that  the  resultant  of  all  the  external 


GENERAL  THEORY  85 

forces  acting  on  a  body  or  system  of  particles  is  equal  to  the  time 
rate  of  change  of  momentum  of  the  system  or  R  =  —  -j-  — 

(a)  Force  Equals  Mass  Times  Acceleration.  —  Let  dR  be  the 
force  exerted  upon  the  elementary  mass  shown  in  Fig.  73.  Let 
the  time  rate  of  flow  be  dm/dt,  where  m  denotes  mass.  Then 
in  an  interval  of  time  dt  there  will  flow  past  any  section  the  mass 
(dm/dt)  dt,  which  will  be  the  amount  considered.  Thus 

dV         dm,\dV      dm  /dV 


Our  discussion  will  be  restricted  to  the  case  where  the  flow  is 
steady  in  which  case  dm/dt  is  constant  and  equal  to  W/g.  There- 
fore, since  (dV/dt)dt  =  dV, 

dR  =  W  dv 

9 

The  summation  of  all  such  forces  along  the  vane  shown  will  give 
the  total  force.  But,  since  integration  is  an  algebraic  and  not  a 
vector  summation  and  in  general  these  various  elementary  forces 
will  not  be  parallel,  it  is  necessary  to  take  components  along  any 
axes.  Thus 


Now  at  point  (1)  the  value  of  Vx  is  Vi  cos  «i  and  at  (2)  it  is  V% 
cos  <*2.  Inserting  these  limits  and  noting  from  Fig.  72  that 
Vo  cos  «2  —  V\  cos  on  =  &VX,  we  have 

W  W 

Rx  =     -  (Vz  cos  «2  —  Vi  cos  «i)  =  —  AFs 
9  0 

(6)  Force  Equals  Time  Rate  of  Change  of  Momentum.  —  Consider 
the  filament  of  a  stream  in  Fig.  74  which  is  between  two  cross- 
sections  M  and  N  at  the  beginning  of  a  time  interval  dt,  and 
between  the  cross-sections  M'  and  N'  at  the  end  of  the  interval. 
Denote  by  dsi  and  ds2  the  distances  moved  by  particles  at  M 
and  N  respectively.  Let  A\  be  the  cross-section  area  at  M, 
Vi  the  velocity  of  the  particles,  and  «i  the  angle  between  the 
direction  of  V\  and  any  convenient  x  axis.  Let  the  same  letters 
with  subscript  (2)  apply  at  N. 

At  the  beginning  of  the  interval  the  momentum  of  the  portion 
of  the  filament  under  consideration  is  the  sum  of  the  momentum 


86  HYDRAULIC  TURBINES 

of  the  part  between  M  and  M'  and  that  of  the  part  between 
M'  and  N.  At  the  end  of  the  interval  its  momentum  is  the  sum 
of  the  momentum  of  the  part  between  M'  and  N  and  that  of  the 
part  between  N  and  N'.  In  the  case  of  steady  flow  and  with  the 
vane  at  rest  or  moving  with  a  uniform  velocity  in  a  straight 
line,  the  momentum  of  the  part  between  M'  and  N  is  constant. 
Hence  the  change  of  momentum  is  the  difference  between  the 
momentum  of  the  part  between  N  and  N'  and  that  of  the  part 
between  M  and  M'.  Noting  that  wAidsi  =  wA2ds2)  since  the 
flow  is  steady,  the  change  in  the  x  component  of  the  momentum 
during  dt  is  then 

/TT 

(  F  2  COS  «2   —    V'l  COS  Ofi). 


FIG.  74. 

If  the  rate  of  flow  be  denoted  by  W  (Ib.  per  sec.),  then 

wAldsl  =  Wdt 

and  the  time  rate  of  change  of  the  x  component  of  the  momentum 
is 

W 

(Vz  COS  <*  2   —    Vi  COS  «i). 

u 

Thus  the  x  component  of  the  resultant  force  is 

W  W 

Rx  =       (Vz  cos  0:2  —  Vi  cos  «i)  =  —  AFX. 

9  9 

This  method  has  the  advantage  that  it  may  be  extended  to 
the  case  where  the  flow  is  unsteady,  if  desired.  In  this  event 
the  two  masses  at  the  ends  would  be  unequal  and  the  momentum 
of  the  portion  from  M'  to  N  would  be  variable.  In  the  case 
of  a  series  of  vanes  on  a  rotating  wheel  running  at  a  uniform 
angular  velocity  the  momentum  of  the  water  on  any  one  vane 


GENERAL  THEORY  87 

will  be  changing.  But  for  the  wheel  as  a  whole,  the  momentum 
of  the  water  on  all  the  vanes  will  be  constant  so  long  as  the  flow 
is  steady. 

The  method  in  (a)  pictures  the  total  force  as  the  vector  sum  of 
all  the  elementary  forces  along  the  path  of  the  stream.  The 
method  in  (b)  shows  that  the  total  force  is  independent  of  the 
path  and  depends  solely  upon  the  initial  and  terminal  conditions. 

Since  the  force  exerted  by  the  water  upon  the  object  is  equal 
and  opposite  to  R,  we  have 

W  W 

Fx  =  ™-  (Fi  cos  ai  -  F2  cos  «2)  =  -         AF.          (13) 

a  ^  __' , .,_— L--      A/.  .    J-T.-     -x ' 

In  a  similar  manner  the  y  component  of  F  will  be 

W  W 

Fy  =  —  (Fi  sin  ai  -  F2  sin  «2)  =  -  •  —  AF,  (14) 

\J  .  yf 

Since  .F  =  VK^'+'/V  and  AF  ==  x/AFg2  +  AF,2, 
"the  value  of  the  resultant  force  Is 

W 

F  =  ~  AF  (15) 


The  direction  of  R  will  be  the  same  as  that  of  AF  and  the  direction 
of  F  will  be  opposite  to  it.  It  is  because  F  and  AT^  are  in  opposite 
directions  that  the  minus  sign  appears 
in  the  last  members  of  equations  (13) 
and  (14).  Note  that  AF  is  the  vector 
difference  between  V\  and  F2. 

58.  Force  upon  Moving  Object. — 
The  force  exerted  by  a  stream  upon 
any  object  may  be  computed  by  the 
equations  of  tfye  preceding  article, 
whether  the  object  is  stationary  or  mov- 
ing. The  principal  difference  is  that  in 

the  latter  case  the  determination  of  AF  may  be  more  difficult. 
Thus  in  Fig.  75  assume  the  initial  velocity  of  the  stream  Fi, 
the  velocity  of  the  object  "ult  the  angle  between  them  ai,  and 
the  shape  of  the  object  to  be  given.  The  relative  velocity  v\ 
can  be  determined  by  the  vector  triangle.  Its  direction  is  also 
determined  by  this  triangle  and  is  not  necessarily  the  same  as 
that  of  the  vane  or  object  struck  by  the  water.  But  the  direc- 
tion of  the  relative  velocity  of  the  water  leaving  is  determined 
by  the  shape  of  the  object,  since  v2  is  tangent  to  the  surface 


88  HYDRAULIC  TURBINES 

at  this  point.  The  magnitude  of  v2  may  be  some  function  of 
»i.  Then  knowing  u2,  which  is  not  necessarily  equal  to  u\  how- 
ever, the  magnitude  and  direction  of  Vz  can  be  computed  from 
the  vector  triangle.  The  AF  desired  is  the  vector  difference  be- 
tween Vi  and  this  F2. 

In  case  the  stream  is  confined  so  that  its  cross-section  is 
known,  the  magnitude  of  vz  may  be  computed  directly  from  the 
equation  of  continuity. 

The  remaining  difficulty  is  the  one  of  determining  the  amount 
of  water  acting  upon  the  body  per  unit  time.  The  rate  of  dis- 
charge in  the  stream  flowing  upon  the  object  is  AiVi  so  that 
W  =  wAiVi.  But  this  may  not  be  the  amount  of  water  striking 
the  object  per  second.  For  instance  if  the  object  is  moving  in 


FIG.  76. — View  showing  action  of  jet  on  several  buckets. 

the  same  direction  as  the  water  and  with  the  same  velocity,  it 
is  clear  that  none  of  the  water  will  strike  it.  The  amount  of 
water  which  will  flow  over  any  object  is  proportional  to  the 
velocity  of  the  water  relative  to  the  object  itself.  If  we  denote 
by  W  the  pounds  of  water  striking  the  moving  object  per  second, 
and  by  a\  the  cross-section  area  normal  to  Vi,  then  W  =  wa\v\. 

If  we  consider  a  wheel  with  a  number  of  vanes  acted  upon  by 
the  water,  the  above  is  true  for  one  vane  only.  The  reason  .that 
less  water  strikes  one  vane  per  second  than  issues  from  the  nozzle 
in  the  same  time  is  that  the  vane  is  moving  away  from  the  nozzle 
and  thus  there  is  an  increasing  volume  of  water  between  the  two. 
But  for  the  wheel  as  a  whole  the  entire  amount  of  water  may  be 
used,  since  one  vane  replaces  another  so  that  the  volume  of  water 


GENERAL  THEORY 


89 


from  the  nozzle  to  the  wheel  remains  constant.  If  one  vane 
uses  less  water  than  is  discharged  from  the  nozzle  in  any  given 
time  interval  and  yet  the  wheel  as  a  whole  uses  the  entire  amount 
of  water,  it  means  that  the  water  must  be  acting  upon  more  than 
one  bucket  at  the  same  instant.  This  is  shown  in  Fig.  76. 

59.  Torque  Exerted. — When  a  stream  flows  through  a  turbine 
runner  in  such  a  way  that  its  distance  from  the  axis  of  rotation 
remains  unchanged,  the  dynamic  force  can  be  computed  from  the 
principles  of  Art.  57.  But  when  the  radius  to  the  stream  varies, 
it  is  not  feasible  to  compute  a  single  resultant  force.  And,  if  it 
were,  it  would  then  be  necessary  to  determine  the  location  of  its 
line  of  action  in  order  to  compute  the  torque  exerted  by  it. 
Hence  we  find  the  total  torque  directly  by  other  means. 


Relative  Path 
of  Water 


Absolute  Path 
of  Water 


A  fundamental  proposition  of  mechanics  is  that  the  time  rate  of 
change  of  the  angular  momentum" (moment  of  momentum)  of 
any  system  of  particles  with  respect  to  any  axis  is  equal  to  the 
torque  of  the  resultant  external  force  on  the  system  with  respect 
to  the  same  axis.1  I  i 

In  Fig.  77  let  MN  represent  a  vane  of  a  wheel  which  may 
rotate  about  an  axis  0  perpendicular  to  the  plane  of  the  paper. 
Water  enters  the  wheel  at  M  and,  since  the  wheel  is  in  motion, 
by  the  time  the  water  arrives  at  N  on  the  vane  that  point  of  the 
vane  will  have  reached  position  N'.  Thus  the  absolute  path 
of  the  water  is  really  MN'. 

1  See  the  author's  "Hydraulics,"  Art.  112.  This  proposition  is  analogous 
to  force  =  time  rate  of  change  of  momentum,  but  here  we  deal  with  moments 
on  both  sides 


90  HYDRAULIC  TURBINES 

Let  us  consider  an  elementary  volume  of  water  forming  a  hollow 
cylinder,  or  a  portion  thereof,  concentric  with  0.  Let  the  time 
rate  of  mass  flow  be  dm/dt.  Then  in  an  interval  of  time  dt, 
there  will  flow  across  any  cylindrical  section  the  mass  (dm/dt)dt. 
Let  this  be  the  mass  of  the  elementary  volume  of  water  we  are 
to  consider.  Let  the  radius  to  this  elementary  cylinder  be  r. 
Only  the  tangential  component  of  the  velocity  will  appear  in  a 
moment  equation,  hence  the  angular  momentum  of  this  cylinder 
of  water  will  be  mass  X  radius  X  tangential  velocity  or  (dm/dt) 
dt  X  r  X  V  cos  a,  and  the  time  rate  of  change  of  momentum, 
which  is  equal  to  torque,  will  be 

d(rV  cos  a) 


In  the  case  of  steady  flow  (dm/dt)  is  constant  and  equal  to  W/g 
and 

W  C2 

Tf  =  -          d(rVcosa). 


Integrating  between  limits  we  have  the  value  of  the  torque 
exerted  by  the  wheel  upon  the  water,  or  by  changing  signs,  the 
value  of  the  torque  T  exerted  by  the  water  upon  the  wheel. 
Thus 

W 

T  =  —  (rlVl  cos  ai  -  r2V2  cos  «2)  (16) 

Representing  the  tangential*  component  of  the  velocity  of  the 
water,  often  called  the  "  velocity  of  whirl/'  by  FM,  since  it  is  in 
the  direction  of  u,  we  have 

T  =  ^  (fi-7.,  -  r2rM2)  (17) 

It  is  immaterial  in  the  application  of  this  formula  whether  the 
water  flows  radially  inward,  as  in  Fig.  76,  radially  outward,  or 
remains  at  a  constant  distance  from  the  axis.  In  any  event  r\ 
is  the  radius  at  entrance  and  r2  is  that  at  exit. 

A  shorter  method  of  proving  the  above  is  analogous  to  method 
(b)  of  Art.  57.  During  an  interval  of  time  dt  the  wheel  has 
received  angular  momentum  at  M  of  dmriVi  cos  ai  and  given 
up  angular  momentum  at  N'  of  dmr2V2  cos  a2,  assuming  the 
flow  to  be  st<  ady.  And,  since  dm  =  (W/g)dt  for  steady  flow, 
the  time  ral,c  of  change  of  angular  momentum  is  (W/g)(riVi  cos 
ai  —  r2Vz  cos  0-2). 


GENERAL  THEORY  91 

It  is  also  possible  to  consider  Fx  of  equation  (13)  to  be  made 
up  of  two  forces  concentrated  at  the  points  of  entrance  and 
exit.  The  former  is  (W/g)Vi  cos  «i  at  radius  ri  and  the  latter  is 
(W/g)Vz  cos  «2  at  radius  r2.  Taking  the  moments  of  these  two 
tangential  forces,  we  get  equation  (16)  at  once. 

60.  Power  and  Head  Delivered  to  Runner.  —  If  the  flow  is 
steady  and  the  speed  of  the  wheel  uniform,  an  expression  for  the 
power  developed  by  the  water  is  readily  obtained.  From  Art.  56 

Power  =  Wh"  =  7\o. 
Using  the  value  of  T  given  by  (16)  and  noting  that  no  ==  u, 

W 

Power  =  Wh"  =  —  (ftiVi  cos  «i  —  w2F2  cos  t*i)        (18) 
y 

This  is  the  power  actually  developed  on  the  runner  by  the 
water.  It  is  analogous  to  the  indicated  power  of  a  steam  engine. 
The  power  output  of  the  turbine  is  less  than  this  by  an  amount 
equal  to  the  friction  of  the  bearings  and  other  mechanical  losses, 
such  as  windage  or  the  disk  friction  of  a  runner  in  water  in  the 
clearance  spaces. 

Eliminating  the  W  from  the  equation  above  we  have  the  head 
actually  utilized  by  the  runner.  Thus 

h"  =  ehh  =  l  (UlVul  -  u,Vu,)  (19) 

y 

As  just  seen,  the  hydraulic  efficiency  is  equal  to  h"  '/h.     The  net 
head  h  supplied  to  the  turbine  is  used  up  in  the  following  ways  : 


Of  these  items  h"  is  the  head  converted  into  mechanical  work,  the 
second  term  represents  the  energy  dissipated  in  the  form  of  heat 
due  to  internal  friction  and  eddy  losses  within  the  runner,  the 
_third_term  is  the  kinetic  energy  lost  at  discharge,  and  the  fourth 
term  represents  the  loss  in  the  nozzle  of  a  Pelton  wheel  or  the 
case  and  guided  of  a  reaction  turbine.  The  factor  m  in  the 
above  may  be  unity  in  the  case  of  an  impulse  wheel  or  a  reaction 
turbine  without  a  diverging  or  proper  draft  tube.  For  a  reac- 
tion tur&ine  with  an  efficient  draft  tube  it  will  be  less  than  unity. 
61.  Equation  of  Energy  for  Relative  Motion.  —  Using  the 
value  of  h"  given  by  (19)  in  (5)  we  have 


(*  +     •+  5)  - 


h> 


92 


HYDRAULIC  TURBINES 


All  of  the  absolute  velocities  will  be  replaced  in  terms  of  relative 
velocities  as  follows: 


COS  0i 
COS  02 


S]  =  Wj  + 
«2  =  W2  + 


COS  0 
COS  0 


The  substitution  of  these  values  gives  us 


,2   _ 


This  equation  serves  to  establish  a  relation  between  points 
(1)  and  (2).  If  the  wheel  is  at  rest  u\  and  w2  become  zero,  Vi  and 
vz  become  absolute  velocities  and  equation  (21)  becomes  the  equa- 
tion of  energy  in  its  usual  form  as  in  (4). 

62.  Impulse  Turbine. — The  following  numerical  solution  is 
given  to  illustrate  the  application  of  the  foregoing  principles. 
This  impulse  turbine  is  of  the  outward  flow  type  known  as  the 


Absolute  Path 
of  Water 


Guide 


Shaft  •    Runner 

FIG.  78. — Outward  flow  turbine. 


Exit 


Girard  turbine.  Obviously  the  direction  of  flow  makes  no  dif- 
ference in  the. theory. 

By  construction,  on  =  18°,  02  =  165°,  n  =  2.0  ft.,  rz  =  2.5  ft. 
The  hydraulic  friction  loss  in  flow  through  the  runner  will  be 
taken  as  proportional  to  the  square  of  the  relative  velocity  so  that 

hr  =  k  ~~-j  where  k  is  an  empirical  constant.     Assume  k  =  0.4. 

Suppose  h  =  350  ft.,  N  =  260  r.p.m.,  q  =  100  cu.  ft.  per  second. 
Find  relative  velocity  at  entrance  to  runner,  relative  velocity 
and  magnitude  and  direction  of  absolute  velocity  at  exit  from 
runner,  head  utilized  by  wheel,  hydraulic  efficiency,  losses,  and 
the  horsepower.  (See  Fig.  77.) 


GENERAL  THEORY  93 

Vi  =  V2gh  =  8.025 V350  =  150  ft.  per  second. 
HI  =  2irri  N/6Q  =  54.4  ft.  per  second. 
U2  =  (r2/ri)ui  =  68.0ft.  per  second. 

By  trigonometry  Vi  =  99.55ft.  per  second. 

Suppose  the  flow  is  in  a  horizontal  plane  so  that  z\  =  z2.     Since 
it  is  an  impulse  turbine  the  pressure  throughout  the  runner  will 
be  atmospheric.     Thus  pi  =  p2. 
Equation  (21)  then  becomes 


1.4  v2*  =  9910  +  4624  -  2960  =  11,574 

v2  =  90.9  ft.  per  second. 

By  trigonometry  V2  =  30.6  ft.  per  second,  «2  =  130° 

Vui  =  VL  cos  on  -  150  X  0.951  =  143 
Vu2  =  az  +  P2    cos  .&  -  68.0  -  90.9  X  0.966  =  -19.7 
(Also   Vu2  =  V2  cos  a2     30.6  -  X  (-  0.639)  =   -19.7) 

=  ^  (54.4  X  143  +  68  X  19.7;  =  283  ft/ 

Oju .  2i 

Hydraulic  efficiency  =  h"/h  =  283/350  =  0.81. 
Hydraulic  friction  loss  =  k-^-  =  0.4  ^   .  =  51.3  ft. 


y  2 
Discharge  loss  =  -^-  =  ^-^  =  14.6  ft. 

Wh"       100  X  62.5  X  283 
:^50:  -550" 

63.  Reaction  Turbine.1  —  Another  numerical  case  will  be  given 
to  illustrate  the  application  of  the  foregoing  principles  to  the 
reaction  turbine.  The  turbine  used  here  is  the  Fourneyron  or 
outward  flow  type,  though  the  theory  applies  to*  any  type. 

By  construction,  ai  =  18°,  fc  =  165°,  ri  =  2.0  ft.,  ra  =  2.5  ft., 

Ai  =  1.36sq.ft.,a2  =  1.425  sq.ft.    Assumed  =  0.2/ft'  =  fc^-)- 

1  See  Art.  8.  If  the  area  a2  is  made  small  enough  the  wheel  passages  will 
be  completely  filled  with  water  under  pressure.  We  then  have  a  reaction 
turbine.  Note  that 


+     '  so  that  Vi  is  not  equal  to 


94  HYDRAULIC  TURBINES 

Suppose  h  =  350  ft.,  N  =  525  r.p.m.,  and  q  =  164.5  cu.  ft.  per 
second.  Find  head  utilized  by  turbine,  hydraulic  efficiency, 
losses,  pressure  at  guide  outlets  (entrance  to  turbine  runner), 
and  the  horsepower. 

Since  the  wheel  passages  are  completely  filled  the  areas  of  the 
streams,  A  i  and  az  are  known,  thus 


Vl  =  q/Ai  =  164.5/1.36  =  121  ft.  per  second. 
v%  =  (A  i/  a*)  Vi  =  115.5  ft.  per  second. 

For  the  above  r.p.m.  HI  =  110  ft.  per  second,  uz  =  137.5  ft.  per 
second. 

Vui  =  Vi  cos  ai  =  115. 

VU2  =  u2  +  v2  cos  j8a  =  137.5  -  115.5  X  0.966  =  26.0. 

h"  =  -  (UlVul  -  u,Vu2)  =  ^-=  (110  X  115  -  137.5  X  26)  = 
g  OA  .  * 

282  ft. 

Hydraulic  efficiency  =  282/350  =  0.805. 

TT       1  T       f    •    ^  1  7    ^22  rk  n   13350  .     _    ., 

Hydraulic  friction  loss  =  K-^-  =  0.2—       -  =  41.5  ffc. 

2g  64.4 

By  trigonometry  V2  =  41  ft.  per  second. 


Discharge  loss  =          =  =  26  ft. 

Zg  O4.4 

Since  v%  is  determined  by  the  area  a2  we  do  not  have  the  use  for 
equation  (21)  that  we  did  in  the  case  of  the  impulse  turbine. 
By  it,  however,  we  can  compute  the  difference  in  pressure  between 
entrance  to  and  discharge  from  the  runner.  Thus  from  (21), 
taking  zi  =  z2, 


w        w  2g 

(If  the  turbine  discharges  into  the  air  then  —  -  =  0  and   -1  =122 

w  w 

ft.)     This  pressure  difference  may  also  be  computed  from  equa- 
tion (5). 

W  h"       62.5  X  164.5  X  282 


550 


=5270h.p. 


64.  Effect  of  Different  Speeds.  —  If  a  wheel  is  run  at  different 
speeds  under  the  same  head,  the  quantities  vi,  v2)  V2,  a2,  h", 
efficiency,  and  power  all  vary.  In  Fig.  79  may  be  seen  the 
velocity  diagrams  for  entrance  and  discharge  from  a  wheel  at  five 


GENERAL  THEORY  95 

different  speeds  from  standstill  to  run-away  and  photographs1 
of  the  wheel  at  these  speeds.  The  relations  for  a  reaction 
turbine  would  be  very  similar  to  these  for  the  impulse  wheel. 
The  most  important  change  is  that  in  the  quantities  a2  and 
F2.  When  the  wheel  is  at  rest,  a2  =  02  and  F2  ='  t>2.  As  the 
speed  increases  o:2  decreases,  passes  through  90°,  and  approaches 
0°.  The  value  of  F2  decreases  to  a  minimum  and  then  increases 
again. 

From  equation  (20)  it  may  be  seen  that,  other  losses  being 
equal,  the  maximum  efficiency  would  be  obtained  when  the  dis- 
charge loss  is  a  minimum.  It  can  be  seen  that  F2  is  very  small 
when  either  v%  =  uz  or  «2  =  90°.  A  means  of  computing  the 
speed  necessary  for  this  will  be  given  later.  Neither  of  these 
gives  the  actual  mathematical  minimum  but  they  are  very 
close  to  it. 

The  torque  exerted  on  the  wheel  by  the  water  may  be  seen  to 
decrease  as  the  wheel  speed  increases.  In  equation  (17)  W  and 
FMi  are  practically  constant,  though  they  vary  slightly  in  the 
case  of  the  reaction  turbine.  But  Fw2  continuously  increases 
algebraically.  It  has  its  maximum  negative  value  when  the  wheel 
is  at  standstill,  it  is  zero  when  the  speed  is  such  that  «2  =  90°, 
and  it  attains  its  maximum  positive  value  when  the  turbine  is 
at  run-away  speed.  This  is  the  maximum  speed  which  the  wheel 
can  reach  under  a  given  head  and  is  attained  when  all  external 
load  is  removed.  Under  these  circumstances  the  torque  exerted 
by  the  water  is  just  sufficient  to  overcome  the  mechanical  losses 
of  the  turbine.  The  run-away  speed  of  the  wheel  is  thus  strictly 
limited  by  hydraulic  conditions. 

In  the  ideal  case  the  maximum  possible  speed  of  the  Pelton 
wheel  would  be  when  the  velocity  of  the  buckets  equalled  the 
velocity  of  the  jet.  But  under  these  conditions  AF  of  equation 
(15)  would  equal  zero.  Consequently  the  wheel  must  run  at  a 
speed  somewhat  less  than  this  as  some  power  is  required  to  over- 
come the  mechanical  losses  at  this  speed.  For  the  impulse  wheel 
the  maximum  value  of  <£  usually  attained  is  about  0.80  at  run- 

1  These  photos  also  show  the  needle  in  the  center  of  the  jet.  The  piece 
at  the  side  of  the  buckets  toward  the  lower  right  hand  side  of  the  case  is 
the  "stripper,"  its  function  being  to  deflect  water  that  might  otherwise  be 
carried  around  with  the  wheel  up  into  the  upper  part  of  the  case.  The 
buckets  pass  through  this  with  a  relatively  small  clearance.  A  close  inspec- 
tion of  the  views  of  the  wheels  and  the  water  leaving  it  will  give  one  a  fair 
idea  of  the  variation  in  velocities. 


96 


HYDRAULIC  TURBINES 


*          o£ 


g,  a 

CO  O> 

1  I 

I  ~ 


GENERAL  THEORY 


97 


away.  For  the  reaction  turbine  it  is  about  1.3.  The  relations 
in  the  latter  case  are  somewhat  more  complex  but  are  similar 
to  those  for  the  Pelton  type. 

65.  Forced  Vortex. — A  forced  vortex  is  produced  when  a 
liquid  is  compelled  to  rotate  by  means  of  external  forces  applied 
to  it.  Thus,  if  the  vessel  X  Y  of  Fig.  80,  is  rotated  about  the  axis 
0-0,  the  water  filling  the  vessel  will  tend  to  rotate  at  the  same 
speed  with  it  and  we  will  have  a  forced  vortex. 

The  pressure  within  this  body  of  water  will  then  vary  as  shown 
by  the  curve  CD.  The  law  of  variation  may  be  found  as  follows 


10 


I'Af 


FKJ.  80. —  Forced  and  free  vortices. 

Consider  an  elementary  volume,  whose  length  along  the  radius 
is  dr  and  whose  area  normal  to  this  is  dA,  and  which  rotates  at 
an  angular  velocity  co  at  radius  r.  The  difference  in  the  pressures 
on  the  two  faces,  which  is  the  resultant  force  acting,  is  equal  to 
dp  X  dA,  and  the  acceleration  of  the  mass  is  co2r,  directed  toward 
the  axis  of  rotation.  Thus 

dpdA  =  (wdA  dr/g)  o>2r 
dp  =  (wuz/g)  rdr 
p  =  (w  <o2/#)  r2/2  -}-  constant. 

To  find  the  value  of  the  constant  of  integration,  let  p0  be  the  pres- 
sure when  r  equals  zero.  Thus  the  constant  is  equal  to  p0,  and 


98  HYDRAULIC  TURBINES 

p  =  rV2       p0 
w        2g         w 

=  "2  +  V"  (22) 

2g       w 

From  this  it  may  be  seen  that  the  curve  is  a  parabola.  If  the 
vessel  is  open,  but  with  sides  high  enough  so  that  the  water  cannot 
overflow,  the  surface  of  the  water  will  become  a  paraboloid,  since 
the  pressure  variation  along  the  radius  is  the  same  whether  the 
water  be  confined  or  not. 

This  equation  is  really  a  special  case  of  equation  (21)  with  v\ 
and  vz  equal  to  zero,  since  there  is  no  flow  ot  water.  If  water 
flows  then  equation  (21)  must  be  employed.  Flow  may  occur 
in  either  direction.  It  may  be  noted  that  the  energy  of  the  water- 
is  not  constant  along  the  radius,  as  both  the  pressure  and  the 
velocity  of  the  water  increase.  This  is  possible  because,  due  to 
the  action  of  external  forces,  energy  is  being  delivered  to  (or 
taken  from)  the  water. 

An  important  application  of  equations  (21)  and  (22)  is  in  con- 
nection with  the  centrifugal  pump.  The  vessel  XY  of  Fig.  78 
may  be  said  to  be  a  crude  illustration  of  the  impeller  of  such  a 
pump.  But  the  equations  are  also  of  value  in  determining  the 
conditions  of  flow  through  turbine  runners,  of  either  the  impulse 
or  reaction  type. 

66.  Free  Vortex. — A  free  vortex  is  produced  when  a  liquid 
rotates  by  virtue  of  its  own  angular  momentum,  previously  de- 
rived from  some  source,  and  is  free  from  the  action  of  external 
forces.  Thus  in  Fig.  80,  if  the  rotating  vessel  XY  is  surrounded 
by  a  stationary  vessel  MN  into  which  the  water  can  pass  from 
XY,  the  water  will  still  tend  to  rotate  and,  neglecting  friction, 
we  will  have  a  free  vortex. 

The  pressure  within  the  free  vortex  will  vary  as  shown  by  the 
curve  DE.  The  law  of  variation  may  be  found  as  follows:  Since 
no  external  forces  are  applied,  the  resultant  torque  exerted  is 
zero,  and  hence  the  angular  momentum  is  constant  (Art.  59). 
Since  angular  momentum  is  proportional  to  rV  cos  a  or  rVu, 
it  follows  that 

rV  cos  a  =  rVu  =  constant  (23) 

the  value  of  the  constant  being  the  value  obtained  from  the 
numerical  value  of  these  factors  at  the  point  of  entrance. 

Considering  the  radial  component  of  the  velocity  Vr,  we  have 


GENERAL  THEORY  99 

q  =  2wrb  X  Vr  =  constant,  from  the  equation  of  continuity. 
Hence 

rbV  sin  a  =  rbVr  =  constant  (24) 

the  value  of  this  constant  being  proportional  to  q. 

The  total  velocity  V  is  the  resultant  of  these  two  components 

so  that . 

72  =  Fu2  +  V2  (25) 

Since  no  energy  is  delivered  to  (or  taken  from)  the  water,  the 
value  of  the  effective  head  H  must  remain  constant.  Thus 

V2       p 
H  =  z  -f  o     +       =  constant, 

the  value  of  this  constant  being  determined  by  the  total  head  of 
the  water  initially.  Taking  the  datum  plane  such  that  z  =  0, 
we  have 

<n  V2  V    2  V  2 

P  -  77  _  *     -  H  —  --— —  ••  (26) 

w  2g  -  2g         2g 

The  flow  of  the  water  may  be  in  either  direction.  Actual  fric- 
tion losses  will  modify  the  resulting  values  of  the  pressure  and 
also  of  Vu,  but  cannot  alter  Vr:  If  b  is  constant,  Vu  and  Vr  vary 
in  the  same  proportion,  neglecting  friction,  so  that  a  is  constant 
and  the  path  of  the  water  is  the  equi-angular  or  logarithmic 
spiral. 

The  free  vortex  is  found  in  the  casing  surrounding  the  im- 
peller of  some  types  of  centrifugal  pumps.  It  is  also  found  in 
the  water  in  a  spiral  case  approaching  a  turbine  runner,  and 
the  above  equations  have  many  applications. 

For  example  equations  (23),  (24),  and  (25)  show  that  the 
velocity  of  the  water  varies  inversely  as  the  radius  of  curvature 
of  its  path.  Hence  if  the  vanes  of  turbine  runners  are  so 
shaped  that  the  stream  lines  have  sharp  curvatures,  the  velocity 
of  the  water  will  be  excessive  and;  from  equation  (26),  it  may 
be  |seen  that  the  pressure  will  be  -reduced.  This  may  result 
in  the  pressure  becoming  so  low  that  erosion  will  result, 
as  mentioned  in  Art.  45.  For  the  same  reason  it  is  undesirable 
to  let  the  water  discharged  from  a  turbine  runner  flow  direct 
toward  the  axis,  as  in  the  pure  radial  inward  flow  turbine. 
For  if  the  water  leaving  the  runner  had  any  " whirl"  this  would 
increase  as  the  axis  was  approached  and,  according  to  the 
equation,  would  become  infinity,  while  the  pressure  would  be 


100 


HYDRAULIC  TURBINES 


minus  infinity,  when  the  radius  equalled  zero.  While  these 
limits  could  not  be  reached,  the  vaporization  of  the  water  that 
would  actually  take  place  would  induce  corrosion  and  also 
cause  additional  eddy  losses.  Hence  the  water  is  turned  at 
discharge  as  has  been  shown  in  Fig.  34,  page  43,  and  the 
central  space  is  often  taken  up  with  a  cone. 

67.  Theory  of  the  Draft  Tube. — The  flow  of  water  through  a 
draft  tube  is  no  different  in  principle  from  the  flow  through  any 


'1111    ***-- 

-^-^-  *-  -4  -\-     


FIG.  81. 

other  conduit  and  hence  Bernoulli's  theorem,  otherwise  known  as 
the  general  equation  of  energy  of  Art/54,  may  be  applied  to  it. 
Equation  (4)  however  has  been  stated  only  for  the  case  of 
steady  flow  and  for  the  present  purpose  we  are  concerned  with 
any  condition  of  flow  that  may  exist.  Hence  we  shall  add  another 
term  called  the  acceleration  head,  which  is  the  head  necessary  to 
accelerate  the  velocity  of  the  water  when  the  rate  of  discharge 
is*changed  by  the  action  of  the  governor.  Let  this  head  be 
denoted  by  hacc,  while  the  loss  of  head  in  friction,  H',  is  divided 
into  its  two  separate  factors,  hf,  the  friction  loss  in  the  tube,  and 


GENERAL  THEORY  101 

Vs2/2g,  the  velocity  head  lost  at  discharge  from  the  mouth  of  the 
draft  tube.     Thus  referring  to  Fig.  81,  we  have 


0  + 

w 


where  p2  denotes  absolute  pressure  and  pa  denotes  atmospheric 
pressure.     Then 

HZ  —  H±  =  H'  -\-  hacc 

' 


The  solution  of  this  equation  will  give  the  allowable  height  of  the 
turbine  runner  above  the  tail-  water  level.  Or  the  equation  can 
equally  well  be  used  to  determine  the  pressure  for  any  given 
elevation.  In  the  above, 

-  is  governed  by  the  altitude  and  local  variations  but  is 
approximately  equal  to  34  ft.  of  water. 

—  cannot  be  less  than  the  vapor  pressure  of  the  water  at  that 

temperature  as  determined  from  the  steam  tables  and  should  be 
from  at  least  2  to  4  ft.  of  water  more. 

Vz  is  a  function  of  the  design  and  type  of  the  runner.  The 
higher  the  capacity  and  speed  of  the  type  the  higher  will  be  the 
value  of  Vz-  It  is  also  a  function  of  the  head  under  which  the 
turbine  runs,  because  all  velocities  vary  as  the  square  root  of  the 
head.  Also  if  a2  is  not  90°,  the  value  of  Vz  will  be  greater  than 
q  divided  by  draft  tube  area. 

hf  depends  upon  the  construction  of  the  draft  tube.  Ordinarily 
it  may  be  assumed  as  about  15  to  25  per  cent,  of  Vz2/2g. 

F3  is  controlled  by  the  setting  of  the  plant  for  that  fixes  the 
allowable  length  of  the  draft  tube.  It  is  also  a  function  of  the 
construction  of  the  draft  tube  and  the  head  under  which  the 
turbine  operates. 

hacc  is  determined  by  the  action  of  the  governor  and  it  may 
be  either  plus  or  minus  in  value.  The  negative  value  is  the 
one  to  use  in  the  above  equation. 

It  can  be  seen  from  the  foregoing  that  two  types  of  turbines 
with  different  discharge  velocities  would  have  different  limiting 


102  HYDRAULIC  TURBINES 

values  of  22  under  the  same  head.  And  the  same  runner  would 
also  require  a  lower  setting  under  a  higher  head  because  of  the 
change  in  this  same  item. 

If  the  turbine  is  set  higher  than  the  limiting  value,  as  deter- 
mined by  this  equation,  the  efficiency  of  conversion  in  the 
draft  tube  will  be  lost  due  to  the  vaporization  and  subsequent 
recondensation  of  the  water.  Also  corrosion  of  the  runner  will 
take  place  because  of  the  liberation  of  oxygen.  Again  if  the 
height  is  very  close  to  the  allowable  limit  for  steady  flow,  the 
sudden  closure  of  the  turbine  gates  by  the  governor  may  cause  the 
pressure  at  discharge  from  the  runner  to  drop  to  such  a  low  value 
that  the  water  vaporizes.  But  an  instant  later  the  water  will 
surge  back  up  the  draft  tube,  striking  the  runner  a  decided  blow. 

68.  QUESTIONS  AND  PROBLEMS 

1.  How  is  the  effective  head  to  be  measured  on  the  Pelton  wheel  and  on 
the  reaction  turbine?     Why  are  two  values  possible  in  the  latter  case? 
What  are  the  definitions  of  the  various  efficiencies  that  may  be  dealt  with? 

2.  What  is  the  procedure  for  computing  the  force  exerted  by  a  stream 
upon  a  moving  object?     What  are  the  reasons  for  the  difference  between 
W  and  W"! 

3.  What  becomes  of  the  total  energy  supplied  in  the  water  to  the  wheel? 
As  the  speed  of  a  wheel  varies,  under  a  constant  head,  the  torque  exerted 
on  it,  and  consequently  its  power,  varies.     Since  the  power  supplied  in  the 
water  is  constant,  what  becomes  of  the  difference  between  the  two? 

4.  What  are  the  fundamental  differences  between  the  solution  of  the  prob- 
lem of  the  impulse  wheel  and  of  the  reaction  turbine? 

5.  As  the  speed  of  a  wheel  changes  how  do  V%  and  0:2  vary?     Of  what 
significance  is  this?     What  limits  the  maximum  speed  of  a  Pelton  wheel 
under  a  given  head? 

6.  What  is  a  forced  vortex?     How  does  the  pressure  vary  in  it?     What 
examples  of  it  are  found? 

7.  What  is  a  free  vortex?     How  does  the  velocity  vary  in  it?     How  does 
the  pressure  vary?     What  common  examples  of  this  are  found? 

8.  What  conclusions  can  one  draw  from  the  equations  for  the  free  vortex 
that  have  an  important  practical  application? 

9.  Derive  the  equation  for  the  maximum  allowable  height  of  a  turbine 
runner  and  discuss  the  items  that  affect  this  value? 

10.  What  is  the  effect  of  the  action  of  the  governor  upon  the  conditions 
within  the  draft  tube?     What  will  be  the  effect  if  a  turbine  runner  is  set 
too  high? 

11.  In  the  test  of  a  reaction  turbine  the  following  readings  were  taken 
(see  Fig.  71) :     Pressure  at  entrance,  pc/w  =  126.6  ft.,  zc  =  12.6  ft.,  diameter 
at  C  =  30  in.,  diameter  at  D  =  60  in.,  and  rate  of  discharge  =  44.5  cu.  ft. 
per  second.     Compute  the  head  on  the  turbine  by  each  of  the  two  methods 
given.  Ans.     140.5  ft. 


GENERAL  THEORY  103 

12.  A  jet  of  water  2  in.  in  diameter  and  with  a  velocity  of  100  ft.  per  sec- 
ond issues  from  a  nozzle  on  the  end  of  a  6-in.  pipe  and  strikes  a  flat  plate 
normally.     Find:  (a)  Power  of  jet,  (b)  thrust  exerted  on  pipe,  (c)  force  ex- 
erted on  plate.  Ans.     (a)  38.4  h.p.,  (b)  376  lb.,  (c)  423  Ib. 

13.  Suppose  the  jet  in  problem  (12)  strikes  a  vane  which  deflects  it  60° 
without  loss  of  velocity.     Find  (a)  component  of  force  in  direction  of  jet, 
(6)  component  normal  to  jet,  (c)  magnitude  and  direction  of  total  force. 
Solve  also  assuming  the  terminal  velocity  to  be  reduced  to  80  ft.  per  second, 
all  other  factors  remaining  the  same. 

Ans.     (a)  211  lb.,  (b)  365  lb.,  (c)  422  lb.  at  60°  with  Vi. 
(a)  254  lb.,  (6)  293  lb.,  (c)  388  lb.  at  49°  08': 

14.  Solve  problem   (13)  assuming  the  angle  of  deflection  to  be   180°. 
What  difference  does  the  angle  make  in  the  magnitude  of  each  force?     What 
difference  is  there  in  the  effect  of  friction  in  each  instance  ? 

Ans.     (a)  844  lb.,  (b)  0,  (c)  844  lb.  at  0°  with  Vi. 
(6)  760  lb. 

15.  Suppose  the  vane  in  problem  (14)  moves  in  the  same  direction  as  the 
jet  with  a  velocity  u,  and  that  friction  loss  is  such  that  v2  =  O.Svi.     When 
u  has  values  of  0,  30,  44.4,  70  and  100  ft.  per  second,  find:  (a)  Values  of  the 
lb.  of  water  per  second  striking  the  vane,  (6)  values  of  absolute  velocity  at 
discharge,  (c)  values  of  the  force  exerted. 

Ans.     (a)  136.3,  95.3,  75.7,  40.8,  and  0  lb.  per  second. 

(b)  -80, -26,  0,  +46,  and  +100  ft.  per  second. 

(c)  760,  372,  234,  68.5,  and  0  lb. 

16.  Solve  problem  (15)  if  the  jet  is  upon  a  wheel  equipped  with  similar 
vanes.     Find  the  power  delivered  to  the  shaft  at  each  speed.     What  be- 
comes of  the  difference  between  this  and  the  power  of  the  jet? 

Ans.     760,  532,  422,  228,  and  0  lb. 
0,  29.0,  34.0,  29.0,  and  0  h.p. 

17.  For  a  turbine  runner,  Vi  =  70  ft.  per  second,  V-i  =  20  ft.  per  second, 
ri  =  2  ft.,  r2  =  3  ft.,  an  =  20°,  «2  =  80°,  and  W  =  300  lb.  per  second,     (a) 
Find  torque  exerted  upon  the  wheel,     (b)  If  u\  =  50  ft.  per  second,  find 
the  power.  Ans.     (a)  1128  ft.  lb.,  (b)  51.3  h.p. 

18.  For  a  turbine  runner,  Vi  =  70  ft.  per  second,  Vz  —  20  ft.  per  second, 
Pi/w  =  25  f;-.,    pz/w  =  —25  ft.     Assume   friction  loss    (kv-22/2g}   in   flow 
through  runner  as  5.78  ft.  and  that  there,isno  difference  in  elevation,     (a) 
Find  head  utilized  by  runner,  (b)  If  W  =  300  lb.  per  second,  find  the  power. 

Ans.     (a)  94.2  ft.,  (b)  51.3  h.p. 

19.  For  the  impulse  turbine  in  Art.  62  it  will  be  found  that  v2  =  u«  when 
u\  =  68.4  ft.  per  second.     Find  the  r.p.m.,  efficiency,  losses,  and  horse- 
power.    Compare  with  values  given  in  Art.  62. 

Ans.     326  r.p.m.,  e  =  0.845,  3365  h.p. 

20.  For  the  reaction  turbine  in  Art.  63  it  will  be  found  that  «2  =  90°  if 
ui  =  86.3  ft.  per  second.     At  that  speed  the  rate  of  discharge  will  be  found 
(by  method  given  later)  to  be  159  cu.  ft.  per  second.     Find  the  r.p.m., 
efficiency,  losses,  and  horsepower.     Compare  with  values  given  in  Art.  63. 

Ans.     412  r.p.m.,  e  =  0.852,  5380  h.p. 

21.  Compare  the  best  r.p.m.  of  the  impulse  turbine  with  the  best  r.p.m. 
of  the  reaction  turbine  in  Problems  (19)  and  (20).     Compare  the  values  of  v* 


104  HYDRAULIC  TURBINES 

in  Problems  (19)  and  (20).  Why  are  these  different?  What  effect  has  this 
upon  the  best  speed? 

22.  Water  enters  the  spiral  case  of  a  turbine  with  a  velocity  of  10  ft.  per 
second,     (a)  Considering  this  velocity  as  tangential  at  a  radius  of  9  ft., 
which  is  the  distance  from  the  runner  axis  to  the  center  of  the  case  near  the 
point  of  intake,  what  is  the  tangential  component  of  the  velocity  at  entrance 
to  the  speed  ring  vanes,  the  outer  radius  of  which  is  7  ft.?     (6)  If  the  height 
of  the  vanes  at  this  point  is  5  ft.,  find  the  radial  component  of  the  velocity 
if  the  turbine  discharges  900  cu.  ft.  per  second,     (c)  What  should  be  the 
angle  of  the  speed  ring  vanes  at  this  point?     (d)  What  should  be  the  angle 
of  entrance  to  the  turbine  guide  vanes,  if  the  radius  is  6  ft.,  and  the  height 
is  3  ft.?  Ans.     (a)  12.85  ft.  per  second,  (&)  4.09  ft.  per  second. 

23.  A  turbine  running  under  a  head  of  200  ft.  is  of  such  a  design  that 
Vp/Zg  =  7  per  cent,  of  h  and  TV/20  =  1  per  cent,  of  h.     If  the  minimum 
pressure  allowable  is  3  Ib.  per  sq.  in.,  what  is  the  maximum  height  the 
runner  may  be  set  above  the  tail-water  level,  assuming  the  draft  tube  loss 
to  be  25  per  cent,  and  the  maximum  negative  acceleration  head  to  be  50 
per  cent,  of  F22/20?     What  will  be  the  result  if  this  same  turbine  is  used 
under  a  head  of  50  ft.?  Ans.     11.6  ft.,  23.2  ft. 

24.  A  turbine  running  under  a  head  of  50  ft.  is  of  such  a  design  that  Vzz/2g 
=  20  per  cent,  and  V$zf2g  =  2  per  cent,  of  h.     If  the  minimum  pressure 

allowable  is  3  Ib.  per  sq.  in.,  what  is  the  allowable  height  the  runner  may  be 
set  above  tail-water  level,  assuming  the  draft  tube  loss  to  be  25  per  cent, 
of  V^/Zg  and  the  maximum  negative  acceleration  head  to  be  50  per  cent, 
of  F22/2<7?  Compare  with  second  part  of  preceding  problem. 

Ans.     15.6  ft. 


CHAPTER  VIII 
THEORY  OF  THE  TANGENTIAL  WATER  WHEEL 

69.  Introductory. — The  tangential  water  wheel  has  been 
classed  as  an  impulse  turbine  with  approximately  axial  flow. 
The  term  tangential  is  applied  because  the  center  line  of  the  jet 
is  tangent  to  the  path  of  the  centers  of  the  buckets.  In  this 
article  the  assumption  will  therefore  be  made  that  a\  =  0°  and 
that  r1  =  r2.  It  will  be  shown  later  that*these  assumptions  are 
not  entirely  correct. 


vl 

FIG.  82. 

If  the  angle  on  be  assumed  equal  to  zero  then  HI  and  Vi  are 
in  the  same  straight  line  and  Vi  =  Vi  —  Ui.  The  conditions  at 
exit  from  the  buckets  are  shown  in  Fig.  82.  In  applying  equation 
15  we  desire  to  find  only  the  component  of  the  force  tangential  to 
the  wheel  since  that  is  all  that  is  effective  in  producing  rotation. 
Therefore  we  shall  find  only  the  component  of  AF  along  the  direc- 
tion of  Ui.  Thus,  if  F  here  denotes  tangential  force, 

W 

F  =  —  (Vi  -  V2  cos  «2) 

==  -~-  (Fi  —  u2  —  v2  cos  182) 
9 

105 


106  HYDRAULIC  TURBINES 

By  equation  (21)  since  zi  =  z2,  pi  =  p2,  Ui  =  u2,  (1  +  A;)y22 

v\  Vi  —  HI 

~~ 


Substituting  this  value  of  #2  we  obtain 

-  "('  -vrfth  -  '«> 


A  more  exact  value  for  the  force  exerted  may  be  found  in  Art.  72. 
The  above  is  only  an  approximation. 

Multiplying  the  force  given  above  by  the  velocity  of  its  point 
of  application,  we  have  the  power  developed.  Thus 

P  =  Fui  =  W/l_™s^\(Vi     _Mi)Mi  (2g) 

g  \        VI  +  Ay 

The  power  input  to  the  wheel,  including  the  nozzle,  is  Wh,  where 
h  is  determined  as  in  Art.  55.  The  power  in  the  jet  is  WVi2/2g 
and  is  less  than  the  former  by  the  amount  lost  in  friction  in  the 
nozzle.  • 

.  Equation  (28)  is  the  equation  of  a  straight  line.  It  shows  that 
F  is  a  maximum  when  u\  is  zero  and  that  it  decreases  with  the 
speed  until  it  becomes  zero  when  m  =  Vi.  Equation  (29)  is  the 
equation  of  a  parabola.  It  shows  that  the  power  is  zero  when 
Ui  =  0  and  again  when  u\  =  V\.  The  vertex  of  the  curve,  which 
gives  the  maximum  power  and  hence  the  maximum  efficiency,  is 
found  when  u\  =  0.5V  i.  Since  in  reality  both  of  these  curves 
are  altered  somewhat,  when  all  the  factors  are  considered,  some 
of  these  statements  require  modification. 

The  actual  speed  for  the  highest  efficiency  has  been  found  by 
test  to  be  such  that  <j>e  =  0.45  approximately,  while  the  value  of 
the  efficiency  is  about  80  per  cent.  Both  of  these  values  vary 
somewhat  with  the  design  of  the  wheel  and  the  conditions  of  use. 
But  one  can  approximately  compute  the  bucket  speed  and  the 
power  of  any  impulse  wheel,  provided  the  head  and  sue  of 
jet  are  known.  The  bucket  speed  u\  =  </>\/?<7S  while  the 
velocity  of  the  jet  Vi  =  cv\/2g'i.  For  a  good  nozzle  with  full 
opening,  if  equipped  with  a  needle,  the  coefficient  of  velocity 
should  be  about  0.98.  Thus  the  rate  of  discharge  is  determined. 
If  the  diameter  of  the  wheel  is  known,  or  assumed  as  a  function 
of  the  size  of  the  jet,  the  rotative  speed  can  be  computed. 

The  reasons  for  the  modifications  of  the  simple  theory  given 


THEORY  OF  THE  TANGENTIAL  WATER  WHEEL      107 

above  and  an  analysis  oi  the  characteristics  of  an  actual  wheel  are 
given  in  the  following  parts  of  this  chapter. 

70.  The  Angle  a\. — The  angle  ai  is  usually  not  zero  as  can  be 
seen  from  Fig.  83.  One  bucket  will  be  denoted  by  B  and  the 
bucket  j  list  ahead  of  it  by  C.  Different  positions  of  these  buckets 
will  be  denoted  by  suffixes.  The  bucket  enters  the  jet  when  it  is 
at  Bi  and  begins  to  cut  off  the  water  from  the  preceding  bucket  Ci. 
When  the  bucket  reaches  the  position  B2  the  last  drop  of  water 
will  have  been  cut  off  from  C2,  but  there  will  be  left  a  portion  of 
the  jet,  MPX  Y,  still  acting  upon  it.  The  last  drop  of  water  X 
will  have  caught  up  with  this  bucket  when  it  reaches  position  C3. 
Thus  while  the  jet  has  been  striking  it  the  bucket  has  turned 


FIG.  83. 

through  the  angle  BiOCs.  The  average  value  of  ai  will  be  taken 
as  the  angle  obtained  when  the  bucket  occupies  the  mea'n  between 
these  two  extreme  positions.  It  is  evident  that  position  C3  will 
depend  upon  the  speed  of  the  wljeel,  and  that  the  faster  the  wheel 
goes  the  farther  over  will  C3  be.  Thus  the  angle  a  i  decreases  as 
the  speed  of  the  wheel  increases.  The  variation  in  the  value  of 
ai  as  worked  out  for  one  particular  case  is  shown  in  Fig.  85. 

71.  The  Ratio  of  the  Radii. — It  is  usually  assumed  that  n  =  r%. 
However  inspection  of  the  path  of  the  water  in  Fig.  84  (a)  will 
show  that  when  the  bucket  first  enters  the  jet  r2  may  be  less  than 
ri.  When  the  bucket  has  gotten  further  along  r2  may  be  greater 


108 


HYDRAULIC  TURBINES 


than  ri.  The  value  of  x(  =  r2/Vi)  depends  upon  the  design  of  the 
buckets,  and  its  determination  is  a  drafting-board  problem  which 
is  not  within  the  scope  of  this  book.  It  is  evident  that  a  value  of 
%  must  be  a  mean  in  the  same  way  that  a  value  of  a i  is  a  mean. 
And  just  as  on  varies  with  the  speed,  so  also  does  x  vary  with  the 
speed.  A  little  thought  will  show  that  when  the  wheel  is  running 
slowly  compared  with  the  jet  velocity  the  value  of  x  will  be  less 
than  when  the  wheel  is  running  at  a  higher  speed.  This- may  be 
verified  by  actual  observation.  When  the  wheel  is  running  at  its 
proper  speed  it  is  probably  true  that  x  is  very  nearly  equal  to 
unity.  In  any  case  the  variation  of  the  value  of  x  from  unity 
cannot  be  very  great. 


(a)  (6) 

FIG.  84.  —  Radii  for  different  bucket  positions. 

72.  Force  Exerted.  —  The  force  acting  on  the  wheel  may  be 
determined  by  the  principles  of  Art.  58,  but,  if  the  radii  are  qot 
equal  it  will  not  be  convenient  to  use  equation  (15)  on  account  of 
the  difficulty  of  locating  the  line  of  action  of  the  force.  How- 
ever we  can  use  equation  (17)  and  by  it  determine  a  force  at 
the  radius  r\  which  shall  be  the  equivalent  of  the  real  force. 
Dividing  (17)  by  7*1  and  letting  F  denote  tangential  force  we 
obtain 

F  =  -(Vul-xVu,) 


y 


Vui    = 


COS  Oil 


cos 


By  equation  (21) 


By  trigonometry 


i2  -  2ViUi 


THEORY  OF  THE  TANGENTIAL  WATER  WHEEL      109 

Substituting  this  value  of  Vi,  and  with  u2  =  xui, 

(1  +  &>22  =  Fi2  -.2ViUi  cos  ai  +  Ai2. 
Substituting  ^2  from  this  in  the  expression  for  F«2  we  obtain 


TTT"  r 

=  - 


cos  ai  -  zU! 

(30) 

Equation  (30)  is  a  true  expression  for  the  force  exerted.  No 
great  error  is  involved,  however,  by  taking  x  =  1.0.  If  that  is 
done  the  expression  under  the  radical  becomes  the  value  of  v\ 
and  may  be  found  graphically.  For  the  sake  of  simplicity  and 
ease  in  computation  a\  may  be  taken  equal  to  zero  and  the  equa- 
tion then  reduces  to  (28),  but  an  exact  value  of  F  will  not  be  ob- 
tained. There  is  little  excuse  for  taking  k  =  0,  as  most  writers 
do,  for  equation  (28)  is  not  simplified  to  any  extent  and  the  re- 
sults are  entirely  incorrect. 

73.  Power.  —  With  F  as  obtained  from  (30)  the  power  is  given 
by  Fui.     We  may  also  compute  h"  and  obtain  the  power  by 
multiplying  by  W. 

Since  h"  =  -  Uj  (  Vui  —xVu2)  it  is  evident  that  the  expression  for 

«7 

h"  is  the  same  as  (30)  if  Wi  be  substituted  for  W.  Thus  the  expres- 
sion for  power  has  the  same  value  no  matter  from  which  basis  it  is 
derived. 

74.  The  Value  of  W.—  W  is  the  total  weight  of  water  striking 
the  wheel  per  second.     It  is  obvious  that  the  weight  of  water 
discharged  from  the  nozzle  is 

W  =  wAiVi. 

Under  normal  circumstances  all  of  this  water  acts  upon  the  wheel. 
However  for  high  values  of  the  ratio  Ui/Vi  a  certain  portion  of 
the  water  may  go  clear  through  without  having  had  time  to 
catch  up  with  the  bucket  before  the  latter  leaves  the  field  of  action. 
It  is  apparent,  for  instance,  that  if  the  buckets  move  as  fast  as  the 
jet  none  of  the  water  will  strike  them  at  all.  For  all  speeds  less 
than  that  extreme  case  a  portion  of  the  water  only  may  fail  to  act. 
Thus  referring  to  Fig.  83,  it  can  be  seen  that  if  the  wheel  speed  is 
high  enough  compared  to  the  jet  velocity  the  water  at  X  may  not 
have  time  to  catch  up  with  bucket  C.  The  variation  of  W  with 
speed  is  shown  in  a  particular  case  by  Fig.  85. 


110 


HYDRAULIC  TURBINES 


It  may  also  be  seen  that  the  larger  the  jet  compared  to  the 
diameter  of  the  wheel  the 'lower  the  value  of  Ui/Vi  at  which 
this  loss  will  begin  to  occur  and  it  is  not  desired  to  have  it  occur 
until  the  normal  wheel  speed  is  exceeded.  Thus  there  is  a  limit 
to  the  size  of  jet  that  may  be  used  for  a  given  wheel,  as  stated 
in  Art.  30.  For  a  given  diameter  of  wheel,  as  the  size  of  the 
nozzle  is  increased,  larger  buckets  must  be  used  and  they  must 
also  be  spaced  closer  together. 

75.  The  Value  of  k. — The  value  of  k  is  purely  empirical  and 
must  be  determined  by  experiment.  If  the  dimensions  of  the 
wheel  are  known  and  the  mechanical  friction  and  windage  losses 
are  determined  or  estimated,  then  from  the  test  of  the  wheel  the 
horse-power  developed  by  the  water  may  be  obtained.  The 
value  of  k  is  then  the  only  unknown  quantity  and  may  be  solved 


100 


400 


500 


fM)  300 

R.P.M. 
FIG.  85. — Values  of  aL  and  W  for  a  certain  wheel. 

for.  The  value  of  k  is  probably  not  constant  for  all  values  of 
Ui/Vi.  Some  theoretical  considerations,  which  need  not  be  given 
here,  have  indicated  that  it  could  scarcely  be  constant  and  an 
experimental  investigation  has  shown  the  author  that  k  decreased 
as  Ui/Vi  increased.  For  a  given  wheel  speed  however  it  is 
nearly  constant  for  various  needle  settings  unless  the  jet  diameter 
exceeds  the  limit  set  in  Art.  30.  The  crowding  of  the  bucket 
then  increases  the  eddy  losses  and  would  require  a  higher  value  of  k. 

The  value  of  k  may  be  as  high  as  2.0  but  the  usual  range  of 
values  is  from  0.5  to  1.5. 

76.  Constant  Input — Variable  Speed. — The  variation  of  torque 
and  power  with  speed  for  different  needle  settings  is  shown  by 
Fig.  86  and  Fig.  87.  With  the  wheel  at  rest  the  torque  may 


THEORY  OF  THE  TANGENTIAL  WATER  WHEEL      111 

vary  within  certain  limits  as  is  shown  by  the  curve  for  full  nozzle 
opening.  This  is  due  to  differences  in  0:1  and  in  x  for  various 
positions  of  the  buckets.  When  running  at  a  slow  speed  the 
brake  reading  was  observed  to  fluctuate  between  the  limits  shown. 
At  higher  speeds  this  could  not  be  detected.  This  action  is  here 
shown  for  only  one  nozzle  opening  but  it  exists  for  all.  With 
a  given  nozzle  opening  the  horsepower  out  put  is  fixed  and  constant. 
The  horsepower  output  varies  with  the  speed.  It  will  be  noticed 
that  the  maximum  efficiency  is  attained  .at  slightly  higher  speeds 


'0  100  200  300  400 

R.r.M. 

FIG.  86. — Relation  between  torque  and  speed. 


500 


for  the  larger  nozzle  openings  than  Tor  the  smaller.  This  is  due, 
in  part,  to  the  fact  that  the  mechanical  losses,  which  are  practically 
constant  at  any  given  speed,  become  of  less  relative  importance 
as  the  power  output  increases. 

Fig.  88  shows  the  variation  of  the  different  losses  for  a  constant 
power  input  but  a  variable  speed.1 

77.  Best  Speed. — It  is  usually  assumed  that  the  best  speed  is 
the  one  for  which  the  discharge  loss  is  the  least.  As  shown  in  Art. 
64,  the  latter  will  be  approximately  attained  either  when  u2  =  v2 
or  when  «2  =  90°.  In  the  case  of  the  impulse  turbine  the  former 

1  The  curves  shown  in  this  chapter  are  from  the  test  of  a  24-in.  tangential 
water  wheel  by  F.  G.  Switzer  and  the  author. 


112 


HYDRAULIC  TURBINES 


assumption  gives  an  easier  solution.     It  will  be  found  that  HZ  =  v% 
if  u\  is  found  from 

fczW  +  2ViUi  cos  A!  -  Fi2  =  0.* 

An  inspection  of  the  curves  in  Fig.  88  will  show  that  the  highest 
efficiency  is  not  obtained  when  the  discharge  loss  is  the  least.     So 


0  100  200    „  „  ...     300  400  500 

K.l  .M. 

FIG.  87. — Relation  between  power  and  speed  for  different  needle  settings. 


that,  although  the  difference  is  not  great,  the  above  equation 
does  not  give  the  best  speed.     The  hydraulic  friction  losses  and 
*L.  M.  Hoskins,;' Hydraulics,"  Art.  198,  Art.  208. 


THEORY  OF  THE  TANGENTIAL  WATER  WHEEL      113 


the  bearing  friction  and  windage  cause  the  total  losses  to  become 
a  minimum  at  a  slightly  higher  speed.  It  does  not  seem  possible 
to  compute  this  in  any  simple  way  but  it  will  be  found  that  the 
best  speed  is  usually  such  that  ui/V\  =  0.45  to  0.49. 

The  speed  of  any  turbine  is  generally  expressed  as  HI  =  <t>-\/2gh. 
The  coefficient  of  velocity  of  the  nozzle  will  reduce  the*above 


H.P.  at  Nozzle 


0      100      200      300      400      500     600 
R.P.M. 

FIG.  88. — Segregation  of  losses  for  constant  input  and  variable  speed. 

values  slightly,  so  that  the  best  speed  is  usually  such  that 
<t>e  =  0.43  to  0.47 

78.  Constant  Speed — Variable  Input. — The  case  considered  in 
Art.  76  is  valuable  in  showing  us  the  characteristics  of  the  wheel 
but  the  practical  commercial  case  is  the  one  where  the  speed  is 
constant  and  the  input  varies  with  the  load.  From  Fig.  87  it  is 


114 


HYDRAULIC  TURBINES 


500 


400 


100  g 


0       0 


.26  .50  .75  1.00          1.25          1.50          1.75         2.00 

Position  of  .Needle.  Inches 

FIG.  89. — Nozzle  coefficients  and'other  data. 


18 


16 


10  12     13.2 


FIG.  90  —  Relation  of  input   to  output  and  segregation  of  losses  for  variable 
input  and  constant  speed. 


THEORY  OF  THE  TANGENTIAL  WATER  WHEEL      115 

seen  that  the  best  speed  is  275  r.p.m.  That  value  was  taken 
because  the  highest  efficiency  was  obtained  with  the  nozzle  open 
six  turns.  For  that  value  of  N  the  curves  in  Fig.  90  were  plotted. 
It  will  be  noted  that  the  relation  between  input  and  output  is 


100 


100%   =    H.P.   Delivered  to  Nozzle 


10 


12      13,2 


B.H.P. 

FIG.  91. — Efficiencies  and  per  cent,  losses  at  constant  speed. 

very  nearly  a  straight  line.     Above  six  turns  it  bends  up  slightly 
because  the  wheel  is  then  slightly  overloaded. 

The  friction  and  windage  was  determined  by  a  retardation 
run1  and  was  assumed  to  be  constant  at  all  loads.  The  hydraulic 
losses  were  segregated  by  the  theory  already  given  (Art.  62).' 
These  results  plotted  in  per  cent,  are  shown  in  Fig.  91  and  Fig.  92. 

1  See  Art.  101. 


116  HYDRAULIC  TURBINES 

79.  Observations  on  Theory. — The  theory  as  presented  in  this 
chapter  is  of  value  principally  for  the  purpose  of  explaining  the 
actual  characteristics  of  Pel  ton  wheels.  Thus  the  determina- 
tion of  values  of  the  angle  a\  is  a  rather  tedious  process,1  and  it  is 
open  to  question  whether  the  average  value  as  defined  in  Art.  70 
is  really  the  proper  one  to  use.  But  the  important  fact  is  that  the 
angle  is  not  zero  and  that  it  does  vary.  In  similar  manner  the 
determination  of  the  amount  of  water  acting  upon  the  wheel  at 
speeds  above  normal,  and  the  determination  of  the  speed  at  which 
this  waste  of  water  begins,  is  difficult.  But  the  consideration  of 
the  problem  makes  it  clear  why  the  curves  for  the  force  exerted  are 
not  straight  lines,  as  may  be  seen  in  Fig.  86,  and  why  the  right- 
hand  portion  is  steeper.  In  turn  this  explains  why  the  actual 
power  curves  of  Fig.  87  are  distorted  parabolas  with  the  right- 
hand  side  much  steeper  than  the  left-hand  side.  Ideally  the 
maximum  speed  of  the  wheel  should  be  such  that  u}  =  Vi,  but 
actually  the  run-away  speed  is  such  that  <j>  =  0.80  approximately. 
This  is  due  to  the  fact  that  the  proportion  of  the  water  acting  on 
the  wheel  at  higher  speeds  would  become  so  small  that  the  force 
exerted  would  be  less  than  that  required  to  overcome  the  bearing 
friction  and  windage  loss. 

The  losses  computad  by  theory  and  in  part  determined  by 
experiment  are  shown  in  Fig.  88.  If  it  were  not  for  the  waste  of 
water  mentioned  above,  the  discharge  loss  from  the  buckets 
would  be  as  shown  by  the  dotted  line.  Actually  the  loss  of  energy 
in  this  water  is  shown  by  the  solid  curve  to  the  left  of  this,  while 
the  discharge  loss  from  the  buckets  is  only  the  intercept  between 
the  latter  curve  and  the  one  to  its  left. 

The  theory,  as  illustrated  in  Fig.  90,  shows  that  for  a  wheel  at 
the  proper  speed  the  principle  loss  of  energy  is  in  the  buckets. 
This  emphasizes  the  importance  of  close  attention  to  the  proper 
design  of  the  latter.  The  theory  also  shows  that  the  individual 
losses  tend  to  follow  straight  line  laws.  This  means  that  the 
relation  between  input  and  output  is  also  a  straight  line.  When 
the  size  ot  the  jet  becomes  too  large  for  the  particular  wheel,  the 
bucket  losses  increase  more  rapidly  and  hence  the  curve  bends 
upward  at  this  point,  as  shown.  The  relation  between  input  and 
output  is  not  exactly  a  straight  line  for  loads  less  than  that  for 
maximum  efficiency,  but  it  is  nearly  so.  This  is  of  interest  be- 

1  See  "Theory  of  the  Tangential  Water  Wheel,"  by  R.  L,  Daugherty,  in 
Cornell  Civil  Engineer,  Vol,  22,  p.  164  (1914), 


THEORY  OF  THE  TANGENTIAL  WATER  WHEEL      117 

cause,  if  only  a  very  few  points  are  determined  by  test,  the 
complete  curve  can  be  drawn  with  a  reasonable  degree  of  accuracy. 
The  theory  also  shows  that  the  hydraulic  efficiency  of  the  wheel 
alone  is  nearly  constant  from  no-load  to  full-load  at  constant 
speed.  And  considering  the  efficiency  of  the  wheel  and  nozzle 


100  %  =JB"  P.  in  Jet 


10 


0  2  4  6^8  10  12      13.2 

15.  ii.i . 

FIG.  92. — Efficiencies  and  per  cent,  losses  at  constant  speed  based  upon  power 

in  jet. 

together  the  hydraulic  efficiency  does  not  begin  to  drop  off  rap- 
idly until  very  small  nozzle  openings  are  reached.  The  reason 
for  this  is  that  the  vector  velocity  diagrams  upon  which  the 
theory  is  based  are  independent  of  the  size  of  the  jet.  The 
variations  shown  .in  Fig.  92  are  due  to  changes  in  cv  and  k.  This 


118  HYDRAULIC  TURBINES 

is  of  practical  importance  as  showing  why  impulse  wheels  have 
relatively  flat  efficiency  curves. 

80.  Illustrative  Problem. — Referring  to  Fig.  93  let  the  total 
fall  to  the  mouth  of  the  nozzle  be  1000  ft.  Suppose  BC  = 
5000  ft.  of  30-in.  riveted  steel  pipe  and  at  C  a  nozzle  be  placed 
whose  coefficient  of  velocity  =  0.97.  Suppose  the  diameter  of 
the  jet  from  the  nozzle  =  6  in.  Let  this  jet  act  upon  a  tangential 
water  wheel  of  the  following  dimensions:  Diameter  =  6  ft., 
ai  =  12°,  fo  =  170°.  Assume  k  =  0.6,  </>  =  0.465,  and  assume 
bearing  friction  and  windage  =  3  per  cent,  of  power  input  to 
shaft. 

The  problem  of  the  pipe  line  is  a  matter  of  elementary 
hydraulics  and  a  detailed  explanation  will  not  be  given  of  the  steps 
here  employed.  The  coefficient  of  loss  at  B  will  be  taken  as 


FIG.  93. 


1.0,  the  coefficient  of  loss  in  the  pipe  will  be  assumed  0.03.     The 
loss  in  the  nozzle  will  be  given  by(  —  ^  —  Ij  ^-,  where  cv  =  the 

coefficient  of  velocity  and  Vi  the  velocity  of  the  jet.     If  Vc  =  the 
velocity  in  the  pipe  then  the  losses  will  be 


y  2 
Taking  HA  =  1000  ft.  and  HI  =  -—•  then  by  equation  (4)  we 

may  solve  for  ^  =  1.38  ft.  or ~  =  861  ft. 
*g  *g 

Thus  Vc  =  9.42  ft.  per  second  and  Vi  =  235.5  ft.  per  second. 
Rate  of  discharge,  q  =  4.62  cu.  ft.  per  second. 

ff\ 

The  pressure  head  at  nozzle,  — -  =  914.5  ft. 

The    wheel    speed   ut  ^0.465  X  8.025 V915.88  =  113   ft.    per 

second. 

Therefore  N  =  360r.p.m. 


THEORY  OF  THE  TANGENTIAL  WATER  WHEEL      119 

By  methods  illustrated  in  Art.  62,  v\  =  126.7  ft.  per  second, 
v%  =  100    ft.   per  second,   and    Vu2  =  14.5,   assuming  x  =  1.0. 

Thus,  h"  =  —  (vul  -  Fw2  )  =  757  ft. 

The  means  of  obtaining  the  following  answers  will  doubtless 
be  obvious. 

Total  head  available,  HA  =  1000  ft. 

Head  at  nozzle,  Hc  =  915.88  ft. 

Head  in  jet,  #1  =  861  ft. 

Head  utilized  by  wheel,  h"  =  757  ft. 

Total  power  available  at  A  =  5250  h.p. 
Power  at  nozzle  (C)  =  4800  h.p. 

Power  in  jet  =  4520  h.p. 

Power  input  to  shaft  =  3970  h.p. 

Power  output  of  wheel          =  3851  h.p 

Hydraulic  efficiency  of  wheel  =  0.878 

Mechanical  efficiency  of  wheel  =  0.970 

Gross  efficiency  of  wheel  =  0.852 

Efficiency  of  nozzle  =  0.941 
Gross  efficiency  of  wheel  and  nozzle  =  0.801 

Efficiency  of  pipe  line  BC  =  0.915 

Overall  efficiency  of  plant  =  0.733 


81.  QUESTIONS  AND  PROBLEMS 

1.  With  the  simple  theory  of  the  tangential  wheel  what  are  the  relations 
for  torque  and  power  as  functions  of  speed  ?     How  may  the  speed  and  power 
of  an  impulse  wheel  be  computed  in  practice  ? 

2.  What  are  the  true  conditions  of  flow  in  the  Pelton  water  wheel  and 
what  assumptions  are  often  made  in  order  to  simplify  the  theory  ? 

3.  When  may  a  portion  of  the  water  discharged  from  a  nozzle  fail  to  act 
upon    the    wheel?     Why?     What   changes   in   design   will   improve   this 
condition? 

4.  Why  is  the  relation  between  input  and  output  at  a  constant  speed  and 
head  not  a  straight  line  throughout  its  range?     How  does  the  hydraulic 
efficiency  vary  from  no-load  to  full-load  at  constant  speed?     Why? 

5.  Suppose  the  dimensions  of  a  tangential  water  wheel  are:  /32  =  165°, 
<t>  —  0.45,  k  =0.5,  and  the  velocity  coefficient  of  the  nozzle  =  0.98.     If 
the  diameter  of  the  jet  =  8  in.  and  the  head  on  the  nozzle  900  ft.,  compute 
the  value  of  the  force  exerted  on  the  wheel,  assuming  ai  •=  0°  and  x  =  1.0. 

6.  Compute  the  force  on  the  wheel  in  problem  (5)  assuming  a\  =  20°. 


120  HYDRAULIC  TURBINES 

7.  Compute  the  hydraulic  efficiency  of  the  wheel  in  problem  (5).     Is  this 
dependent  upon  the  value  of  the   head? 

8.  Derive  an  equation  for  the  hydraulic  efficiency  of  a  Pelton  wheel, 
giving  the  result  in  terms  of  wheel  dimensions  and  factors  such  as  <f>  and  cr. 

9.  Suppose  it  is  desired  to  develop  2000  h.p.  at  a  head  of  600  ft.     Assum- 
ing an  efficiency  of  600  ft.,  what  will  be  the  size  of  jet  required,  and  what 
will  be  the  approximate  diameter  and  r.p.m.  of  the  wheel? 


CHAPTER  IX 
THEORY  OF  THE  REACTION  TURBINE 

82.  Introductory. — The   main  purpose  of  this  chapter  is  to 
explain  the  characteristics  of  reaction  turbines.    In  turbine  theory 
there  are  many  variables  and  one  must  assume  some  of  these  and 
compute  the  rest,  and,  according  to  what  is  assumed  as  known, 
the  theory  presented  by  various  indididuals  will  differ.     Also 
there  are  matters  of  difference  of  detail.     For  instance  one  may 
assume  the  hydraulic  friction  losses  through  the  entire  turbine, 
including  guides  and  runner,  to  be  some  function  of  the  rate  of 
discharge,  while  another  will  attempt  to  analyze  these  losses  and 
compute  them  individually. 

The  turbine  designer,  desiring  to  obtain  some  definite  perform- 
ance, naturally  assumes  certain  results  and  computes  the  dimen- 
sions necessary.  For  our  present  purpose,  we  shall  do  exactly 
the  opposite  of  this  and  assume  all  the  dimensions  as  known  and 
endeavor  to  determine  the  characteristics  of  the  given  turbine. 

83.  Simple  Theory. — A  very  simple  theory  is  possible  by  as- 
suming certain  factors  to  be  known  as  the  result  of  experience. 
Thus,  as  in  the  case  of  the  impulse  wheel,  the  peripheral  velocity 
of  the  runner  may  be  represented  as  Ui  =  <j>\/2gh.  _And_  the 
speed  at  which  the  efficiency  is  a  maximum  is  given  by  values  of 
<t>e  ranging  from  0.55  to  0.90  according  to  the  type  of  the  runner 
as  in  Fig.  ,34,  page  43.     This  differs  from  the  Pelton  wheel  not 
only  in  the  numerical  values  of  <f>e  but  also  in  the  much  greater 
range  that  is  possible. 

The  efficiency  of  the  turbine  may  be  assumed  as  from  80  to  90 
per  cent,  according  to  the  size  and  type  of  the  runner,  and  hence 
the  power  may  be  computed  if  the  rate  of  discharge  is  known. 
We  here  introduce  another  factor  c  such  that  V\  =  c\/2gh. 
(This  is  really  a  velocity  coefficient  but  there  is  no  need  to  draw 
any  distinction  between  it  and  the  coefficient  of  discharge,  since 
here  the  coefficient  of  contraction  is  unity.)  It  may  be  noted 
that  in  the  reaction  turbine  the  water  is  under  pressure  through- 
out its  flow  and  hence  the  total  energy  of  the  water  entering  the 
runner  is  not  all  kinetic.  Thus  the  coefficient  c  can  never  be  unity.' 
And  again,  since  the  water  flows  through  a  closed  conduit  all 

121 


122  HYDRAULIC  TURBINES 

the  way  from  the  case  to  the  tail  race  it  is  evident  that  any  loss 
of  head  in  any  part  must  cause  a  change  in  the  rate  of  discharge. 
And  since  the  losses  within  the  turbine  runner  vary  with  the  speed, 
it  is  evident  that  the  rate  of  discharge,  and  hence  of  c,  must  also 
vary.  This  again  is  different  from  the  impulse  wheel,  where  the 
action  of  the  wheel  has  no  effect  upon  the  velocity  of  the  water 
from  the  nozzle.  Thus  the  factor  c  is  not  only  less  than  unity 
but  it  depends  upon  the  design  and  type  of  the  runner,  and  fur- 
thermore it  varies  with  the  speed  of  the  latter.  Because  of  this 
variation  with  the  speed,  we  shall  here  give  only  the  values 
obtained  at  speeds  which  result  in  the  highest  efficiency  being 
obtained  from  the  wheel.  In  practice  ce  varies  from  0.6  to  0.8 
according  to  the  type  of  runner.  Then 

q  =  A!  X  ce\/2gh~. 

It  may  be  of  interest  to  note  that  as  one  proceeds  from  runners 
of  Type  I  to  Type  IV  of  Fig.  34,  page  43,  one  gets  farther  away 
from  the  impulse  wheel  in  all  respects.  Not  only  are  the  resulting 
operating  characteristics  and  conditions  of  service  more  unlike 
but  the  numerical  factors  are  of  increasing  difference.  Thus 
values  of  (j>e  for  the  reaction  turbine  are  larger  than  for  the  im- 
pulse turbine  and  they  increase  in  the  direction  mentioned.  The 
pressure  pi  is  zero  for  the  impulse  turbine,  but  not  for  the  reac- 
tion turbine.  For  the  same  head  and  setting,  the  value  of  p\ 
will  increase  from  Type  I  to  Type  IV.  But  if  pi  increases,  V^ 
must  decrease.  Hence  high  values  of  ce  accompany  low  values 
of  <$>e  and  vice  versa. 

For  the  present  we  are  assuming  that  values  of  <f>e  and  ee  are 
to  be  chosen  according  to  the  type  of  runner  concerned. 

84.  Conditions  for  Maximum  Efficiency.— To  obtain  the  best 
efficiency  the  water  must  enter  the  runner  without  shock  and 
leave  with  as  little  velocity  as  possible.  In  order  to  enter  without 
shock  the  vane  angle  must  agree  with  the  angle  j8i  determined  by 
the  velocity  diagram  and,  in  the  case  of  the  reaction  turbine,  the 
velocity  vi  as  determined  by  the  velocity  diagram  should  be  equal 
in  magnitude  to  that  determined  by  the  rate  of  discharge  and  the 
runner  area  01.  In  order  to  leave  with  as  little  velocity  as  pos- 
sible the  angle  <x2  may  be  made  equal  to  90°,  as  has  been  shown  in 
Art.  64.  In  the  early  type  of  turbine  as  built  by  Francis  such  an 
angle  would  make  the  water  flow  along  a  radius  and  hence  such 
a  discharge  was  called  "radial"  discharge.  With  the  develop- 


THEORY  OF  THE  REACTION  TURBINE          123 

ment  of  the  mixed  flow  type,  this  term  is  no  longer  appropriate 
but  it  is  quite  commonly  used  nevertheless.  Such  a  condition  is 
also  spoken  of  as  "  perpendicular  "  discharge,  from  the  fact  that 
the  absolute  velocity  of  the  water  is  normal  to  the  linear  velocity 
of  the  vane,  and  the  term  "  axial"  flow  is  also  usedjrom  the  fact 
that  with  the  high-speed  type  of  runner  the  flow  is  approximately 
parallel  to  the  axis. 

A  further  reason  for  the  use  of  az  =  90°  as  desirable  for  a 
high  efficiency  of  the  read  ion  turbine  is  that  otherwise  the  water 
would  enter  the  draft  tube  with  a  whirling  motion  which  would 
increase  the  losses  within  the  latter. 

85.  Determination  of  Speed  for  Maximum  Efficiency.  —  A 
runner  of  rational  design  would  be  so  proportioned  that  there 
would  be  no  shock  at  entrance  for  the  same  speed  at  which  the 
discharge  velocity  would  be  normal  to  the  vane  velocity. 
That  is  a2  =  90°  and  p\  =  pi  at  the  same  speed,  where  £'1  is 
the  angle  of  the  runner  vane  and  /3i  the  angle  of  Vi  as  determined 
by  the  vector  diagram.  The  following  equations  therefore 
apply  only  to  such  a  runner. 

From  the  velocity  diagrams  we  have,  if  p'i  =  $\. 

Vi  sin  ai  =  Vi  sin  jS'j 
Vi  cos  ai  =  Ui  +  Vi  cos  |8'i 
Eliminating  v\  between  these  two  equations  we  have 

_  sin  (0'j  -  ai)  ,     . 

Ui~   ~^wr    l 

as  the  relation  between  u\  and  V\  when  there  is  no  abrupt  change 
of  velocity  at  entrance  to  the  runner. 

Since  a2  =  90°  and  hence  Vu^  =  0,  we  have  from  equation  (19) 


€kh  =  =  (33) 

9  ff 

as  the  relation  between  HI  and  V\  for  which  the  discharge  loss  is 
a  minimum. 

Solving  equations  (32)  and  (33)  simultaneously  we  have 


jeh2ghrin  (ff'i 
\      2  sin  £'i  cos 


=     /_      e*2gh  sin  jg;i 
'    \2sin  (j8'i  -  «0 


124  HYDRAULIC  TURBINES 

From  this  it  follows  that 


leh  sin  (0'i  -  ai) 
*e  =:  \2sin/J'1coB«1 

/Z        6/t  sin  ff'i  ,«KN 

:  \2  sin  (/5'i  -  «0  cos  <*! 


It  must  be  borne  in  mind  that  the  preceding  equations  apply 
only  to  a  runner  designed  as  stated.  For  any  runner,  whether  of 
rational  design  or  not,  the  value  of  <f>  necessary  to  make  a2  = 
90°  can  be  determined  by  involving  more  dimensions  than  the 
above,  and  such  an  expression  will  now  be  derived.  It  will  be 
assumed  also  that  this  is  the  most  efficient  speed  for  any 
runner,  though  this  may  not  be  strictly  true  if  the  entrance  loss 
is  not  zero  at  this  speed. 

If  0:2  =  90°,  VU2  =  Uz  -f  v  2  cos  02  =  0.  Since  u2  =  xui  and 
vz  =  (A  i/  0,2)  Vi  =  yVi  from  the  equation  of  continuity,  we  may 
write 

x  ui  +  y  Vi  cos  fa  =  0  (36) 

as  the  relation  between  ui  and  V\  for  which  a2  =  90°.     Solving 
this  simultaneously  with  equation  (33)  we  obtain 


eh'2gh-y 
'  \   -  2x  c 


cos  02 


y  I         €h'2gh'X 

\  —  2y  cos  02  cos 
From  this  it  follows  that 


cos 


Ce 


\-22/cos02cosa1  (38) 


From  equation  (33)  we  may  write  Ui  Vi  =  eh  •  2gh/2cos  a\  and 
from  this  it  follows  that 

<t>ece  =  eh/2  cos  on  (39) 

Values  of  eh  and  ai  change  somewhat  with  different  types  of  tur- 
bines but  this  shows  that  the  factors  (f>e  and  ce  vary  approximately 
inversely,  as  stated  in  Art.  83. 

The  equations  of  this  article  are  all  based  upon  assumptions 
which  prescribe  special  relationships  between  c  and  0,  and  are 


THEORY  OF  THE  REACTION  TURBINE          125 

true  only  for  a  special  value  of  </>.  A  method  of  determining  c 
for  any  value  of  <£  will  be  found  in  Art.  87. 

86.  Losses. — The  net  head  supplied  the  turbine  is  used  up  in 
two  ways;  in  hydraulic  losses  and  in  mechanical  work  delivered 
to  the  runner.  The  head  utilized  in  mechanical  work  is  h"  = 

~(uiVUl  —  u2VU2).  In  accordance  with  the  usual  method  in  hy- 
draulics we  may  represent  hydraulic  friction  loss  in  the  runner  by 
k  v22/2g,  k  being  an  experimental  constant.  If  the  turbine  dis- 
charges into  the  air  or  directly  into  the  tail  race  the  discharge 
loss  is  V^/2g.  In  addition  there  may  be  a  shock  loss  at  en- 
trance to  the  runner.  The  term  shock  is  commonly  applied  here 
but  the  phenomena  are  rather  those  of  violent  turbulence.  This 
turbulent  vortex  motion  causes  a  large  internal  friction  or  eddy 
loss. 

Referring  to  Fig.  94,  the  value  of  vi  and  its  direction  are  de- 
termined by  the  vectors  ui,  and  V\.  Since  the  wheel  passages 


1 


77 


c  c 

FIG.  94. 

are  filled  in  the  reaction  turbine,  the  relative  velocity  just  after 
the  water  enters  the  runner  is  determined  by  the  area  «i  and  its 
direction  by  the  angle  of  the  wheel  vanes  at  that  point.  If  all 
loss  is  to  be  avoided,  these  values  should  agree  with  those  deter- 
mined by  the  vector  diagram;  but  that  is  possible  for  only  one 
value  of  Ui  for  a  given  head.  For  any  other  condition  the  velocity 
Vi  at  angle  ft  is  forced  to  become  v'\  at  angle  0Y  This  causes  a 
loss  of  head  which  will  be  assumed  to  be  equal  to  (CC')2/2g. 
Since  the  area  of  the  stationary  guide  outlets  normal  to  the 
radius  should  equal  the  area  of  the  wheel  passages  at  entrance 
normal  to  the  radius,  the  normal  component  (i.e.,  perpendicular 
to  ui)  of  vi,  should  equal  that  of  V\.  Therefore  CCf  is  parallel 
to  ui  and  its  value  is  easily  found  to  be 

nn,  _             sin  (|8']  —  «i)  T7 
uo    —  u\  —  -  — -. — —, K  i 


126  HYDRAULIC  TURBINES 

If  k'  =  sin  (0'i  —  «i)/  sin  0'i,  then 

shock  loss  =  -    — » — 

87.  Relation  between  Speed  and  Discharge. — Equating  the 
net  head  to  the  sum  of  all  these  items  we  have 


T,"**  j.  I?2 
k+ 


2g 


, 


2(u,F.1  - 


All  velocities  can  be  expressed  in  terms  of  u\  and  V\  as  follows  : 
HZ  =  xu\,  v2  =  yVi,  Vui  =  Vi  cos  on, 
Vuz  =  u2  +  v2  cos  02  =  a?wi  +  2/F]'  cos  02; 


cos  32  = 


cos 


i.o 


0        0.1      0.2       0.3      0.4      0.5      0.6       0.7       0.8      0.9       1.0      1.1       1.2     1.8     1.4 

Values  of   p 

Peripheral  Speed,  u  \  =  ^>  ~\l    %  gh 

FIG.  95. — Comparison  of  the  relation  between  c  and  0  as  determined  by  theory 

and  by  test. 

Making  these  substitutions  and  reducing  we  obtain, 
[(1  +  k)y2  +  fc'*]Fi2  +  2(cos  «,.  -  fc')FiWi  +  (1  -  a;2)^]2=  2^. 

From  this  equation  Fi  may  be  computed  for  any  value  of  wheel 
speed,  MI.     It  is  customary  to  express  the  wheel  speed  as  ui  = 
<j>-\/2gh,  and  we  may  also  say  Vi  =  c\/2gh.     The  use  of  these 
factors  is  more  convenient  in  general.     Introducing  them  our 
equation  becomes 

f(l  +  fc)2/2+fc'2]c2+2(cos  ai-fcOc^+Cl-s2)^!.          (40) 
From  this  equation  c  may  be  computed  for  any  value  of  <£.     For 


THEORY  OF  THE  REACTION  TURBINE         127 

the  outward  flow  turbine  (37)  becomes  a  hyperbola  concave  up- 
ward, for  the  inward  flow  turbine  it  becomes  an  ellipse  concave 
downward. 

A  comparison  between  the  values  of  c  as  determined  by  this 
equation  and  as  determined  by  experiment  is  shown  in  Fig.  95. 
One  turbine  was  an  outward  flow  turbine  and  the  other  was  a 
radial  inward  flow  turbine.  Considering  the  imperfections  and 
limitations  of  the  theory,  the  agreement  is  remarkably  close. 

If  the  turbine  discharges  into  an  efficient  draft  tube  the  discharge 
loss  may  be  represented  by  mV^/Zg,  where  m  is  a  factor  less  than 
unity.  If  there  were  no  internal  friction  and  eddy  looses  within 
the  tube,  the  value  of  m  would  depend  only  upon  the  areas  of  ends 
of  the  tube  and  would  be  equal  to  (Az/Az)2.  Actually  m  is 
greater  than  this  due  to  hydraulic  friction  losses.  And  as  the 
speed  of  the  turbine  departs  from  the  normal  value,  it  is  probable 
that  m  increases  still  more  and  approaches  unity.  Introducing 
the  discharge  loss  as  mV^/2g  in  the  equation  at  the  beginning  of 
this  article,  we  obtain  as  a  substitute  for  equation  (40), 

[(m  +  k)y2  +  fc'2]c2  +  2[cos  ai  -  k'  -  (1  -  m)xy  cos  fc]c0 

-  [(2  -  m)x2  -  1]02  =  1  (41) 

It  will  be  found  that  this  equation  will  give  slightly  higher  values 
of  c  than  equation  (40),  which  is  to  be  expected.  Thus  the  use  of 
a  diverging  draft  tube  increases  the  power  of  the  turbine  not  only 
by  increasing  its  efficiency  but  also  by  increasing  the  quantity  of 
water  it  can  discharge. 

If  desired,  equation  (36),  when  put  in  terms  of  0  and  c,  can  be 
solved  simultaneously  with  equations  (40)  or  (41),  thus  giving  a 
third  method  of  computing  the  value  of  <j>e.  Also  it  is  possible  to 
derive  a  general  equation  for  the  efficiency  of  a  reaction  turbine 
and  by  calculus  find  the  value  of  <f>  for  which  the  efficiency  is  a 
maximum.  However  the  resulting  equation  is  somewhat  lengthy 
and,  because  it  is  of  no  practical  value,  will  not  be  given  here. 
Values  of  0  determined  by  it  will  usually  not  differ  much  from 
those  determined  by  the  simpler  approximate  method  of  assuming 
that  <*2  =  90°. 

88.  Torque,  Power  and  Efficiency. — General  equations  for 
torque,  power,  and  efficiency  were  derived  in  Chapter  VII  and  the 
application  of  these  illustrated  by  a  numerical  case  in  Art.  63. 
In  that  article  the  speed  and  rate  of  discharge  of  the  turbine  were 
assumed.  In  the  present  chapter  methods  are  shown  for  com- 


128  HYDRAULIC  TURBINES 

puting  by  theory  the  speed  for  maximum  efficiency  and  the  rate 
of  discharge  for  any  speed.  From  this  point  on  the  procedure 
is  the  same  as  in  Chapter  VII. 

It  is  of  course  possible  to  make  algebraic  solutions  for  these 
quantities  and  the  resulting  equations  then  express  results  in 
terms  of  known  factors  and  dimensions.  Thus,  to  illustrate,  the 
hydraulic  efficiency  is  in  general 


th  =  h"  '/h  =  (uiVi  cos  «i  —  U2V2  cos  a^/gh. 

For  the  reaction  turbine  in  particular  u2  =  xui  and  F2  cos  a2  = 
u2  +  v2  cos  /32  =  xui  +  yVi  cos  /?2.  Substituting  in  the  above  we 
obtain  e^  =  [(cos  a\  —  xy  cos  j32)  V\u\  —  x2Ui2]/gh.  From  this  it 
follows  that 

eh  =  2(cos  ai  -  xy  cos  /32)  c<j>  -    2#202  (42) 

A  numerical  result  in  a  given  case  can  be  computed  either  by 
substituting  the  known  quantities  in  the  above  equation  or  by 
computing  the  separate  items  of  the  general  equation.  The  latter 
usually  involves  no  more  labor. 

Equation  (42)  is  of  interest  because  it  involves  no  arbitrary 
factors  of  loss.  Thus  if  the  relation  between  speed  and  discharge 
is  known,  as  by  experiment,  the  hydraulic  efficiency  can  be 
computed,  provided  the  proper  wheel  dimensions  are  known. 
Actually  it  is  so  difficult,  as  will  be  explained  later,  to  determine 
the  proper  values  of  the  runner  dimensions,  that  the  numerical 
accuracy  of  the  result  is  doubtful.  The  hydraulic  efficiency  can 
probably  be  estimated  more  accurately  than  the  separate  factors 
in  these  equations.  The  euqation  is  of  very  practical  value 
however  in  showing  that  the  hydraulic  efficiency  is  independent 
of  the  head  under  which  the  turbine  is  run. 

Equation  (42)  is  perfectly  general  for  any  reaction  turbine 
and  is  not  restricted  to  the  maximum  efficiency.  The  value  of  the 
maximum  efficiency  will  be  obtained  by  using  the  values  of  <t>e  and 
ce  in  the  equation.  Of  course,  since  Vu  is  assumed  to  be  zero  for 
this  case  the  value  of  the  maximum  efficiency  can  be  computed 
much  more  directly  than  by  the  use  of  the  above. 

89.  Variable  Speed  —  Constant  Gate  Opening.  —  Since  c  varies 
with  the  speed  the  input  for  a  fixed  gate  opening  will  not  be  con- 
stant for  all  speeds  as  it  is  in  the  case  of  the  impulse  turbine.  The 
variation  of  the  losses  at  full  gate  with  the  speed  ranging  from  zero 
up  to  its  maximum  value  is  shown  by  Fig.  96.  The  horse-power 


THEORY  OF  THE  REACTION  TURBINE         129 

in  each  case  obtained  by  multiplying  wq/55Q  by  the  head  lost  as 
given  by  Art.  86. 

The  curves  for  the  impulse  turbine  in  Fig.  88  may  also  represent 
percentages  by  the  use  of  a  proper  scale  since  the  input  is  con- 
stant. But  for  the  reaction  turbine  the  percentage  curves  will 
be  slightly  different  from  those  in  Fig.  96.  It  will  thus  be  true 
that  the  speed  at  which  the  efficiency  is  a  maximum  will  be  slightly 
different  from  the  speed  for  which  the  power  is  the  greatest. 

The  Francis  turbine  for  which  the  curves  in  Fig.  95  and  Fig.  96 
were  constructed  had  the  following  dimensions : 


Francis  Turbine  at  Full  Gate 

:  1  Ft  Head  x  h  3/2        Peripheral  Speed 


0        .1        .2        .3        .4        .5        .6         .7         .8        .9        1.0       1.1  -    1.2     1.3 
Values  of  <? 

FIG.  96. — Losses  at  full  gate  and  variable  speed. 


=  13°,  0'!  =  115° 
.ft., 


2  =  165°,  Ai  =  5.87  sq.  ft.,  a2  =  6.83  sq. 
=  4.67  ft.,  r2  =  3.99  ft. 


From  this  data  x  =  0.855,  y  =  0.860,  k'  =  1.08,  and  k  =  0.5  (as- 
sumed). Attention  is  called  to  the  fact  that  the  horse-power 
output  was  determined  by  an  actual  brake  test  while  the  horse- 
power input  to  shaft  was  computed  from  the  theory  given  in  the 
preceding  article.  The  two  differ  by  the  amount  of  power  con- 
sumed in  bearing  friction  and  other  mechanical  losses. 

90.  Constant  Speed—  Variable  Input.  —  The  relation  between 
input  and  output  and  the  segregation  of  losses  for  a  cylinder  gate 
turbine  at  constant  speed  is  shown  in  Fig.  97.  In  the  four  tests 
the  gate  was  raised  %,  %,  %,  and  %  of  its  opening.  With 
the  turbine  running  on  full  gate  but  at  an  incorrect  speed  there  is 


130 


HYDRAULIC  TURBINES 


a  shock  loss  at  entrance  as  shown  in  Art.  86.  This  loss  is  due  to 
an  abrupt  change  in  the  direction  of  the  relative  velocity  of  the 
water.  When  the  turbine  is  running  at  the  normal  speed  but 
with  the  gate  partially  closed  there  is  a  shock  loss  of  a  slightly 
different  nature.  A  partial  closure  of  the  gates  increases  the 
value  of  Vi  and  the  angle  on  may  be  affected  somewhat.  How- 
ever q  will  be  reduced  while  ai  remains  the  same  and  thus  the 


3.53 


1.0  2.0 

B.H.P.  under  1  Ft.  Head 

FIG.  97. — Losses  for  cylinder  gate  Francis  turbine  at  constant  speed. 

velocity  Vi  must  be  suddenly  reduced  to  v\.  The  loss  of  head  due 
to  this  may  be  roughly  represented  by  (vi  —  v'i)2/2g.  While  this 
expression  may  not  give  the  exact  value  of  the  loss,  yet  it  must 
be  true  that  it  will  be  of  the  nature  shown  by  the  curves. 

This  loss  is  known  to  be  less  in  the  case  of  the  swing  gate 
turbine  than  in  the  cylinder  gate  turbine.  While  both  on  and  V\ 
are  altered  in  the  case  of  the  wicket  gate,  the  transition  in  the 
runner  is  less  abrupt  and  consequently  the  eddy  losses  are  less. 

There  will  always  be  a  slight  leakage  through  the  clearance 


THEORY  OF  THE  REACTION  TURBINE 


131 


spaces  and  such  a  loss  is  indicated  in  Fig.  97  though  it  was  not 
possible  to  compute  it  with  any  exactness.  It  was  merely  added 
to  show  that  it  exists  but  it  was  not  accounted  for  in  Fig.  96. 
These  results  on  a  percentage  basis  are  shown  in  Fig.  98. 

The  cylinder  gate  turbine  is  rather  inefficient  on  light  loads  due 
to  the  big  shock  loss.  The  wicket  gates  do  not  occasion  such 
large  shock  losses  and  hence  reduce  the  input  curve  to  a  line  more 
nearly  parallel  to  the  output  line  in  Fig.  97,  and  thus  improve  the 
part  load  efficiency  of  the  turbine.  Also  the  cylinder  gate  turbine 
gives  its  best  efficiency  when  the  gate  is  completely  raised  and  the 


100 


1.0  2.0 

B.H.P.  under  1  Ft.  Head 
FIG.  98. — Cylinder  gate  turbine  at  constant  speed. 

power  output  has  its  greatest  value.  But  the  wicket  gate  tur- 
bine usually  develops  its  best  efficiency  before  the  gates  are  fully 
open.  There  is  thus  left  some  overload  capacity. 

91.  Runner  Discharge  Conditions. — The  following  theory, 
though  open  to  certain  objections,  serves  to  explain  the  observed 
phenomena  at  the  discharge  from  a  turbine  runner.  A  low 
specific  speed  type  of  runner,  such  as  Type  I  in  Fig.  34,  page  43, 
will  have  stream  lines  through  it  which  differ  but  little  from  one 
another,  while  a  high  specific  speed  type  such  as  Fig.  99  will 
have  stream  lines  which  differ  considerably.  Thus  in  Fig.  99 
stream  line  (a)  next  to  the  crown  will  have  smaller  radii  at  both 
entrance  and  outflow  than  stream  line  (c)  next  to  the  band. 


132 


HYDRAULIC  TURBINES 


Also  the  radii  of  points  along  the  outflow  edge  vary  considerably 
more  in  the  case  of  the  high  specific  speed  runner.  The  radius  of 
curvature  of  stream  line  (a)  is  much  greater  than  that  of  line  (c) 
and  in  accordance  with  equation  (23)  the  velocity  along  line  (a) 
will  be  less  than  along  line  (c).  It  must  be  noted  that  the  lines 
shown  in  the  drawing  are  really  "  circular"  projections  of  the 
actual  stream  lines,  by  which  is  meant  that  the  various  points  are 
revolved  about  the  axis  of  the  runner  until  they  lie  in  the  plane  of 
the  paper;  and  also  a  free  vortex,  while  existing  in  the  space  be- 

tween guides  and  runner,  and 
also  in  the  draft  tube,  does  not 
exist  within  the  runner.  But, 
considering  the  velocity  com- 
ponent in  the  plane  of  the  paper 
only,  and  considering  the  rota- 
tion about  the  centers  of  curva- 
ture of  the  lines  drawn  (and 
not  about  the  axis  of  the  run- 
ner), the  equations  of  Art.  66 
may  be  applied. 

Since  eh  =  h"/h  =  (uiVui  — 
we  may  write 

~  gh"     (43) 


FIG.  99.  riot   only  for  the  turbine  as  a 

whole,  but  for  each  individual 

stream  line.  It  has  been  stated  that  usually  a  runner  is  so  designed 
that  <*2  =  90°.  With  some  runners  observation  shows  that  there 
is  a  slight  whirl  of  the  water  across  the  entire  draft  tube  at  the 
point  of  maximum  efficiency,  but  this  might  be  expected,  since  the 
assumption  that  Vuz  should  equal  zero  is  a  mere  approximation. 
With  the  low  specific  speed  turbine  it  is  possible  to  have  a2  =  90° 
for  all  stream  lines  at  some  speed  which  may  or  may  not  be  exactly 
the  most  efficient,  but  it  is  difficult  to  do  this  with  the  high 
specific  speed  runner  and  still  satisfy  the  equation  above.  Thus 
suppose  the  discharge  is  normal  to  u2  for  stream  line  (6)  in  Fig. 
99.  Then  for  this  stream  line  UiVu\  =  gh".  Considering  line 
(c)  both  u\  and  Vui  are  larger  for  the  reasons  stated  in  the  first 
paragraph.  If  Vuz  is  to  be  zero  here  also,  h"  must  be  larger. 
But  the  conditions  here  are  not  favorable  to  as  high  an  efficiency 
as  along  line  (b),  because  of  the  proximity  to  the  boundary  (which 


THEORY  OF  THE  REACTION  TURBINE 


133 


in  this  instance  is  the  band)  and  because  of  the  sharper  curvature. 
Hence  by  no  proportioning  can  the  water  be  compelled  to  flow 
as  desired.  Since  UiVui  is  larger  and  In"  is  smaller  than  for  line 
(6)  it  follows  that  the  right  hand  member  of  the  equation  must  be 
positive  and  hence  there  must  be  some  whirl  at  the  point  of  dis- 
charge in  the  direction  of  rotation  of  the  runner.  In  similar 
fashion  there  may  be  a  negative  whirl  at  the  point  of  discharge 
from  line  (a),  but  since  u\,  Vui,  and  h"  all  decrease  for  this  line, 
as  compared  with  (b) ,  it  is  possible  that  there  may  be  little  or  no 
whirl  here.  All  this  reasoning  has  been  verified  by  experimental 
observations.1  This  whirl  of  the  water  near  the  band  decreases 
the  efficiency  of  the  draft  tubes  as  constructed  in  the  past  and 
points  the  reason  for  the  development  of  a  new  type  of  tube  if 
turbines  of  higher  specific  speed  are  to  be  used.  And  unless  more 
effective  draft  tubes  are  used,  this  shows  that  this  factor  tends  to 
reduce  the  efficiency  of  the  runner  as  the  specific  speed  increases. 


FIG.  100. 

For  the  higher  the  specific-  speed  the  greater  the  variation  in  r2 
for  stream  lines  (a)  and  (c),  and  this  has  been  shown  to  be  undesir- 
able. 

At  part  gate  on  any  turbine  the  efficiency  and  hence  h"  are 
known  to  be  less  than  on  full  load,  the  latter  being  taken  as  the 
load  for  which  the  efficiency  is  a  maximum.  And  if  wicket  gates 
are  used  the  angle  <*i  is  less  than  at  full  load  so  that  Vi  cos  a\ 
would  appear  to  be  higher.  Hence  if  the  right  hand  side  of  equa- 
tion (43)  is  equal  to  zero  at  full  load,  it  would  have  a  relatively 
large  positive  value  for  a  partial  opening  of  the  turbine  gates. 
Thus  Vuz  would  be  positive,  which  agrees  with  the  vector  dia- 
gram, since  with  a  smaller  rate  of  discharge  the  velocity  v2  would 
be  less  while  the  wheel  speed  u2  is  considered  to  be  the  same. 

Since  u2  Vu2  must  have  a  large  positive  value  for  a  small  gate 
opening  and  u2  varies  with  the  radius,  it  follows  that  Vu2  is  rela- 
tively small  near  the  band  and  relatively  large  near  the  crown 

1  See  Trans.  A.  S.  C.  E.,  Vol.  LXVI,  p.  378  (1910). 


134  HYDRAULIC  TURBINES 

Since  Vu2  =  ^2  +  v 2  cos  /32  and  w2  is  fixed,  the  value  of  v2  must 
decrease  as  the  crown  is  approached  (cos  /32  is  negative)  and  may 
even  become  negative.  This  means  that  water  is  actually 
pumped  back  into  the  runner  near  the  crown  and  out  near  the 
band.  This  shows  the  undesirability  of  a  large  variation  in  the 
radii  of  the  discharge  edge  of  a  runner,  and  this  latter  is  character- 
istic of  the  profile  of  the  high  specific  speed  runner.  Hence  this 
theory  presents  a  reason  why  the  part  gate  efficiency  of  a  reac- 
tion turbine  must  be  less  as  the  specific  speed  increases. 

The  smaller  the  rate  of  discharge  for  a  given  head  the  less  the 
value  of  vz  at  any  point  on  the  outflow  edge  and  hence  the  less 
the  wheel  speed  necessary  to  make  «2  =  90°  at  this  point.  Thus 
with  any  turbine  the  speed  for  which  the  efficiency  is  a  maxi- 
mum decreases  as  the  gate  opening  decreases. 

92.  Limitations  of  Theory. — The  defects  of  this  theory  or 
any  theory  are  as  follows:  In  order  to  apply  mathematics  in 
any  simple  way  it  is  necessary  to  idealize  the  conditions  of  flow 
by  assuming  that  all  the  particles  of  water  at  any  section  move 
in  the  same  direction  and  with  the  same  velocity.  Such  is 
very  far  from  being  the  case  so  what  we  use  in  our  equations 
is  the  average  direction  and  the  average  velocity  of  all  the 
particles  of  water.  That  in  itself  could  easily  cause  a  discrep- 
ancy between  our  theory  and  the  fact,  because  the  theory  is 
incomplete. 

But  even  to  determine  accurately  these  average  values  that 
are  used  in  the  equations  is  a  matter  involving  some  difficulty. 
Thus,  though  the  direction  of  the  streams  leaving  the  runner  is 
influenced  by  the  vane  angle  at  that  point,  it  cannot  be  said  that 
the  angle  £2  is  exactly  equal  to  the  vane  angle  at  exit.  In  fact 
the  author  has  roughly  proved  by  study  of  a  test  where  some 
special  readings  were  observed  that  the  two  may  differ  by  from 
5  to  10  degrees,  and  that  /32  varied  regularly  for  different  values 
of  <£.  The  same  thing  may  be  said  about  the  area  a2.  Some 
recent  experiments  in  Germany1  have  shown  that  there  may  be  a 
certain  amount  of  contraction  of  the  streams  and  that  this  con- 
traction varies  for  different  speeds.  Thus  the  true  value  of 
a2  may  be  slightly  less  than  the  area  of  the  wheel  passages. 
These  observations  concerning  02  and  a2  apply  equally  well  to 
other  angles  and  areas. 

i  Zeitschrift  des  Vereins  deutscher  Ingenieure,  May  13,  1911. 


THEORY  OF  THE  REACTION  TURBINE 


135 


In  computing  the  results  plotted  in  Fig.  95  the  coefficients  of  c 
and  <£  in  equation  (40)  were  treated  as  constant.  It  has  just 
been  shown  that  the  real  values  of  the  angles  and  areas  may  vary 
slightly  with  the  speed.  Also  it  is  stated  in  Art.  75  that  the  value 
of  k  is  not  constant  at  all  speeds  for  the  impulse  turbine.  While 
the  conditions  with  the  reaction  turbine  are  very  different,  yet 
it  is  doubtless  true  that  k  is  not  strictly  constant  here.  If  it  were 
known  just  how  k  and  the  dimensions  used  varied  with  the  speed, 
the  theoretical  curve  could  be  made  to  more  nearly  coincide  with 
the  actual  curve.  In  addition  the  expression  for  shock  loss  is 
only  an  approximation.  But  even  as  it  is  the  discrepancy  is  not 
serious. 

By  the  use  of  the  proper  average  dimensions  the  equations 
given  may  be  successfully  applied  to  a  radial  flow  turbine.  For 
the  mixed  flow  turbine  they  will  apply  approximately.  The 


Runner 


FIG.   101. 

reasons  for  this  are  that  with  the  mixed  flow  turbine  the  radius  r2 
varies  through  such  a  wide  range  of  value  that  it  is  difficult  to 
fix  a  proper  mean  value;  likewise  the  vane  angle  at  exit  and  also 
the  area  varies  so  radically  that  a  mean  value  can  scarcely  be 
obtained  with  any  accuracy.  Even  if  these  mean  values  could 
be  obtained  the  theory  would  still  be  imperfect,  for  the  reason 
stated  in  the  first  paragraph  of  this  article. 

The  value  of  the  angle  a\  may  be  taken  as  that  of  the  angle 
shown  in  Fig.  101,  though  it  may  be  seen  that  this  is  a  mean  for 
the  various  stream  lines.  The  velocity  of  the  water  through  the 
guide  vanes  may  be  denoted  by  F0  but,  since  the  space  between 
guides  and  runner  is  a  free  vortex,  the  velocity  Vi  is  increased  in 


136  HYDRAULIC  TURBINES 

the  ratio  r0/ri.  In  all  the  discussion  so  far  it  has  been  inferred 
that  the  point  (1)  coincided  with  the  outer  radius  of  the  runner. 
It  is  the  velocity  of  this  point  that  is  given  by  the  factor  <f>  and  most 
of  our  empirical  computations  will  be  based  upon  this,  for  ease 
in  computation.  But  if  one  wishes  to  compute  certain  results 
by  applying  the  laws  of  hydrodynamics,  such  as  equations  (40), 
(41),  and  (42)  for  example,  it  is  desirable  to  select  the  point  (1) 
such  that  the  most  suitable  mean  values  for  use  in  the  equations 
will  be  obtained.1  Such  a  point  is  often  said  to  be  the  center 
of  the  circle  shown  at  entrance  to  the  runner  of  Fig.  101.  The 
diameter  of  this  circle  is  the  shortest  line  that  can  be  drawn  from 
the  tip  of  one  vane  to  the  next  vane. 

A  similar  procedure  should  be  followed  for  the  point  of  dis- 
charge if  the  turbine  were  a  pure  rad'al  flow  turbine  with  all 
points  on  the  outflow  edge  at  the  same  radius.  For  the  mixed 
flow  type  of  turbine  it  can  be  proven  that  the  discharge  may  be 
considered  as  concentrated  at  the  center  of  gravity  of  the  out- 
flow area. 

Because  of  the  difficulties  of  applying  the  theory  in  a  definite 
case  numerical  results  are  of  doubtfuraccuracy.  But  the  theory 
has  other  uses.  Thus  the  theory  shows  why  certain  factors  and 
dimensions  must  vary  with  the  specific  speed  of  the  runner. 
It  shows  that  the  rate  of  discharge  cannot  be  constant  for  a  given 
runner  at  different  speeds  under  a  constant  head  and  gate  open- 
ing. It  shows  why  certain  conditions  are  desirable  for  efficiency 
and  how  the  proper  speed  may  be  approximately  computed.  It 
explains  the  losses  within  a  turbine  and  shows  why  certain 
characteristics  vary  as  they  do.  It  serves  to  give  the  reasons 
why  there  are  fundamental  differences  in  the  operating  charac- 
teristics of  turbines  of  different  types.  In  other  words  it  will 
in  general  furnish  the  reason  for  any  result  found  in  practice. 
And  beyond  explaining  these  characteristics,  it  indicates  the 
effect  of  any  change  in  any  direction. 

93.  Effect  of  y. — The  ratio  of  Ai/a2  is  expressed  by  y.  If  y 
is  small  enough  the  turbine  will  be  an  impulse  turbine,  the  value 
of  <f>  giving  the  best  speed  will  be  about  0.45,  pi/w  =  0,  and  c 
will  equal  1.00  if  the  slight  loss  in  the  nozzle  is  neglected.  (Ac- 

i  This  procedure  was  not  followed  however  in  dealing  with  the  Francis 
turbine  for  which  the  curves  in  this  chapter  were  drawn.  But  this  turbine 
is  much  more  amenable  to  mathematical  analysis  than  the  runners  of  the 
present  day. 


THEORY  OF  THE  REACTION  TURBINE 


137 


tually  c  will  be  the  coefficient  of  velocity  of  the  nozzle  and  will  be 
about  0.97).  As  the  value  of  y  increases,  the  turbine  becomes  a 
reaction  turbine,  the  value  of  <£  increases,  p\/w  increases,  and  c 
decreases.  The  general  tendency  of  these  factors  is  shown  in 
Fig.  102. 

It  is  thus  seen  that  the  design  of  a  reaction  turbine  can  be 
varied  so  as  to  secure  quite  a  range  of  results. 


Values  of  |f    (-  A1 
FlQ.   102. 


1.2 


94.  QUESTIONS  AND  PROBLEMS 

1.  Given  the  diameter  D  and  the  height  B  of  a  turbine  runner,  how  can 
one  approximately  compute  the  speed  and  power  for  any  head? 

2.  Why  does  the  rate  of  discharge  from  a  turbine  runner  vary  with  the 
speed  under  a  fixed  head?     Why  is  the  velocity  of  the  water  entering  the 
runner  less  than  V2g7i? 

3.  What  are  the  conditions  necessary  for  high  efficiency  of  a  reaction 
turbine?     What  effect  does  the  draft  tube  have  upon  this  also? 

4.  In  what  two  ways  may  <f>6  be  computed?     What  are  the  fundamental 
differences  involved  in  these  methods?     Should  the  numerical  results  differ? 

5.  What  are  the  various  losses  of  the  turbine  and  how  may  they  be  ex- 
pressed?    What  is  the  effect  of  the  draft  tube  in  this? 

6.  How  may  a  general  equation  between  speed,  discharge,  and  head  be 
derived? 

7.  How  may  a  general  equation  for  the  hydraulic  efficiency  of  a  reaction 
turbine  be  derived?     What  does  it  indicate? 

8.  From  the  curves,  what  are  the  differences  between  the  variations  in 
the  losses  for  impulse  and  reaction  turbines? 


138  HYDRAULIC  TURBINES 

9.  Explain  why  the  discharge  conditions  for  a  high  specific  speed  runner 
are  less  favorable  than  those  for  a  low  specific  speed  runner  both  being 
assumed  to  be  running  at  their  points  of  maximum  efficiency. 

10.  Explain  why  the  discharge  conditions  at  part  gate  are  less  favorable 
for  the  high  specific  speed  runner  than  for  the  low  specific  speed  runner. 

11.  What  are  the  limitations  of  turbine  theory  and  why?     What  is  the 
value  of  the  theory? 

12.  What  effect  does  the  change  in  the  ratio  of  the  area  through  the  guide 
vanes  to  that  at  outflow  from  the  runner  have  upon  the  values  of  0,  c, 
and  pi/w! 

13.  A  turbine  runner  36  in.  in  diameter  and  12  in.  high  at  entrance  will 
run  at  what  probable  r.p.m.  and  develop  what  power  under  a  head  of  60  ft.? 
(Assume  value  of  «i.)  Ans.     N  =  317,  B.h.p  =  890. 

14.  In  problem   (13)  suppose  the  intake  to  the  runner  is  at  a  height  of 
15  ft.  above  the  tail-water  level.     What  is  the  probable  value  of  the  pressure 
head  at  this  point?  Ans.     29  ft. 

15.  The  dimensions  of  the  original  Francis  runner  were  on  =  13°,  ft\  = 
115°,  ft,  =  165°,  A!  =  5.87  sq.  ft.,  a2  =  6.83  sq.  ft.,  n  =  4.67  ft.,  and  r2  = 
3.99  ft.     Compute  the  values  of  <£e  and  ce  by  the  first  method  given,  assum- 
ing eh  =  0.83.     Do  these  answers  give  shockless  entrance?     Do  they  give 
a2  =  90°?     What  dimensions  could  be  changed  to  make  both  of  these  con- 
ditions, be  fulfilled  at  the  speed  computed?     Ans.     <j>e  =  0.678,  ce  =  0.628. 

16.  ^Compute  the  values  of  <f>e  and  ce  for  the  Francis  turbine  in  the  pre- 
ceding problem  by  the  second  method  given?     Do  these  answers  give  shock- 
less    entrance?     Do   they    give    «2  =  90°?     What   dimensions    could   be 
changed  so  as  to  fulfill  both  these  conditions  at  this  speed? 

Ans.     <j>e  =  0.643,  ce  =  0.663. 

17.  Francis  noted  that  his  runner  was  not  quite  properly  designed  and 
that  there  was  some  shock  loss  at  entrance  when  running  at  the  most  effi- 
cient speed.     By  test  the  actual  value  of  4>e  was  found  to  be  0.67.     Compute 
the  corresponding  value  of  ce  and  compare  with  the  curve  in  Fig.  95.     As- 
sume k  =  0.5.  Ans.     ce  =  0.655. 

18.  Compute  the  hydraulic  efficiency  of  the  Francis  turbine  of  problem 
(15)  using  the  values  of  <f>  and  c  given  in  problem  (17)  and  compare  with 
value  given  by  curve  in  Fig.  98.  Ans.     0.825. 

19.  If  this  turbine  discharges  into  a  draft  tube  of  such  dimensions  that 
m  may  be  assumed  equal  to  0.3,  compute  the  value  of  c  for  a  value  of  <f>e 
equal  to  0.675.    Compute  the  hydraulic  efficiency.    The  value  of  <J>e  has  been 
increased  slightly  here  because  of  the  presumption  that  the  draft  tube  will 
increase  the  efficiency  of  the  turbine.     Compare  with  problems  (17)  and  (18). 

Ans.     ce  =  0.66,  eh  =  0.833. 

20.  What  is  the  percentage  value  of  the  discharge  loss  from  the  Francis 
turbine  of  problem  (15),  assuming  «2  =  90°  and  ce  =  0.66?     For  this  par- 
ticular turbine,  what  is  the  possible  gain  in  efficiency  due  to  using  a  draft 
tube  which  would  reduce  the  velocity  to  zero  without  loss  of  energy?     (Note. 
F2  sin  «2  =  Vz  for  90°  and  F2  sin  «2  =  v2  sin  /32  =  ycV2gh  sin  /32.) 

Ans.     2.15  per  cent. 

21.  If  the  turbine  in'problem  (17)  is  used  under  a  head  of  30  ft.,  find  the 


THEORY  OF  THE  REACTION  TURBINE         139 

r.p.m.,  the  quantity  of  water  discharged,  and  the  power  delivered  to  the 

shaft.     Find  similar  results  for  problem  (19). 

Ans.     N  =  60.2,  q  =  169,  h.p.  =  475,  N  =  60.8,  q  =  170.2,  h.p.    =  484. 

22.  If  the  turbine  in  the  preceding  problem  were  to  be  run  at  the  same 
speed  of  60.2  r.p.m.,  while  the  head  decreased  to  18  ft.,  find  the  rate  of  dis- 
charge, hydraulic  efficiency  and  power. 

Ans.     <f>  =  0.864,  c  =  0.633  by  (40),  q  =  126  5,  eh  =  0.755,  195  h.p. 


CHAPTER  X 


TURBINE  TESTING 

95.  Importance. — Testing  is  necessary  to  accompany  theory 
in  order  that  the  latter  may  be  perfected  until  it  becomes  reliable 
enough  to  be  useful.  Unless  the  theory  agrees  with  the  facts  it 
is  not  true  theory  but  only  an  incorrect  hypothesis.  Only  by 
means  of  theory  and  testing  working  hand  in  hand  can  improve- 
ments in  design  be  readily  brought  about.  Thus  the  ease  of 
testing  is  a  measure  of  the  rate  of  development  of  any  machine. 

Again,  if  we  are  to  thoroughly  understand  turbines,  it  will  be 
necessary  to  make  a  thorough  study  of  test  data  in  order  to  appre- 
ciate the  differences  between  different  types.  Unfortunately 
there  is  a  scarcity  of  good  and  thorough  test  results. 

The  only  public  testing  flume  in  the  United  States  is  the  one  at 
Holyoke,  Mass.  Nearly  3000  runners  ha\  _,  oeen  tested  there  and 
it  has  been  an  important  factor  in  the  development  of  modern 
turbines.  The  maximum  head  obtainable  there  is  about  17  ft., 
also  it  is  scarcely  possible  to  test  runners  above  42  in.  in  diameter 
because  of  the  limitations  imposed  by  the  depth  of  the  flume. 

An  acceptance  test  should  always  be  made  when  a  turbine  is 
purchased  if  it  is  possible  to  do  so.  Otherwise  the  purchaser  will 
have  no  assurance  that  the  specifications  have  been  fulfilled. 
Thus  a  case  may  be  cited  where  the  power  and  efficiency  of  a 
tangential  water  wheel  were  both  below  that  guaranteed  as  can 
be  seen  by  the  following: 


Efficiency 

Normal  h.p. 

Maximum  h.p. 

Guarantee  

0  800 

3500 

5225 

Test.  ..'.  

0  720 

2300 

3500 

In  this  table  the  normal  horse-power  means  the  power  at  which 
the  maximum  efficiency  is  obtained,  any  excess  power  over  that 
being  regarded  as  an  overload.  The  actual  efficiency  is  8  per 
cent,  less  than  that  guaranteed  and  the  wheel  is* really  a  2300- 
h.p.  wheel  instead  of  a  3500-h.p.  wheel.  It  is 'true  that  the 

140 


TURBINE  TESTING 


141 


wheel  could  deliver  3500  h.p.  but  at  an  efficiency  of  only  67 
per  cent.  Since  that  is  the  maximum  power  the  5225-h.p.  over- 
load could  not  be  attained. 

Another  case  may  also  be  given  where  the  facts  are  of  a  differ- 
ent nature.1  A  comparison  of  the  guaranteed  and  test  results 
for  a  reaction  turbine  is  shown  in  -Fig.  103.  The  efficiency  se- 
cured was  higher  than  that  guaranteed,  but  it  was  also  attained  at 
a  much  higher  horse-power.  If  the  turbine  were  then  run  on  the 
load  specified  it  would  be  operating  on  part  gate  all  the  time  and 
at  a  correspondingly  low  efficiency.  This  is  a  common  failing 
in  "cut  and  try"  practice.  A  turbine  of  excess  capacity  is 


provided;  it  never  lies  down  under  any  load  put  upon  it  and  the 
owner  is  satisfied.  Quite  frequently  also  a  turbine  which  must 
run  at  a  certain  speed  is  really  adapted  for  a  far  different  speed. 
Thus  under  the  given  conditions  its  efficiency  may  be  very  low, 
when  the  runner  might  really  be  excellent  if  operated  at  its  proper 
speed.  A  test  would  show  up  these  defects,  otherwise  they  may 
remain  unknown. 

Another  reason  for  making  tests  would  be  to  determine  the 
condition  of  the  turbine  after  length  of  service.  The  effect  of 
seven  years'  continuous  operation  upon  a  certain  tangential  water 
wheel  is  seen  in  Fig.  104.  This  drop  in  efficiency  is  due  to  rough- 
ening of  the  buckets,  to  wear  of  the  nozzle,  and  to  the  fact  that 
end  play  of  the  shaft  together  with  the  worn  nozzle  caused  the 
jet  to  strike  upon  one  side  of  the  buckets  rather  than  fairly  in  the 
center.  It  might  be  noted  however  that  a  7-ft.  wheel  of  the 
same  make  in  the  same  plant  showed  no  change  in  efficiency  after 

1  Trans.  A.  S.  C.  E.,  Vol.  LXVI,  p.  357. 


142 


HYDRAULIC  TURBINES 


the  same  length  of  service.  With  reaction  turbines  the  guides 
and  vanes  become  worn  and  the  clearance  spaces  also  wear  so  as  to 
permit  the  leakage  loss  to  increase. 

As  to  whether  efficiency  is  important  or  not  depends  upon  cir- 
cumstances. If  there  is  an  abundance  of  water  in  excess  of  the 
demand  the  only  requirement  is  that  the  turbine  deliver  the  power 
demanded.  But  where  the  water  must  be  purchased,  as  it  is  in 
some  cases,  or  where  vast  storage  reservoirs  are  constructed  at 
considerable  expense  it  is  desirable  that  water  be  used  with  the 
utmost  economy. 


70 


20 


10 


1&05 


0  10  20  30  40  50  GO 

B.H.P. 

FIG.  104. — Effects  of  service  upon  a  42"  tangential  water  wheel. 


96.  Purpose  of  Test. —  The  nature  of  the  test  will  depend  upon 
the  purpose  for  which  it  is  made.  In  a  general  way  there  are  four 
purposes  as  follows : 

1 .  To  Find  Results  for  Particular  Specified  Conditions.     This  will 
usually  be  an  acceptance  test  to  see  if  certain  guarantees  have 
been  fulfilled.  •  The  guarantee  will  usually  specify  certain  values 
of  efficiency  obtained  at  certain  loads  at  a  fixed  speed  under  a 
given  head.     Occasionally  several  values  of  the  head  will  be 
specified. 

2.  To  Find  Best  Conditions  of  Operation.     Such  a  test  will  cover 
a  limited  range  of  speed,  load,  and  head;  all  of  them,  however, 
being  in  the  neighborhood  of  the  maximum  efficiency  point. 
A  test  of  this  nature  will  show  what  a  given  turbine  is  best 
fitted  for. 


TURBINE  TESTING  143 

3.  To  Determine  General  Principles  of  Operation. — This  test  is 
similar  to  the  above  except  that  it  is  more  thorough.      It  should 
cover  all  speeds  from  zero  to  the  maximum  possible  under  no  load. 
Various  gate  openings  may  be  used  and  the  head  may  also  be 
varied.     Such  a  test  will  enable  one  to  understand  the  turbine 
better  and  could  also  be  used  to  verify  the  theory. 

4.  To  Investigate  Losses. — This  test  would  be  similar  to  the 
preceding  except  that  a  number  of  secondary  readings  of  veloc- 
ities, pressures,  etc.,  at  various  points  might  be  taken.     Such  a 
test  will  be  of  interest  chiefly  to  the  designer. 

97.  Measurement  of  Head. — The  head  should  be  measured  as 
close  to  the  wheel  as  possible  in  order  to  eliminate  pipe-line  losses. 
The  head  to  be  used  should  be  as  specified  in  either  equation  (7) 
or  (8)  or  (9)  of  Art.  55,  according  to  circumstances.     The  pressure 
may  be  read  by  means  of  a  pressure  gage  if  it  is  high  enough. 
For  lower  heads,  a  mercury  column  or  a  water  column  will  give 
more  accurate  results.     Care  should  be  taken  in  making  connec- 
tions for  the  pressure  reading  so  that  the  true  pressure  may  be 
obtained.     The  reading  of  any  piezometer  tube  will  be  correct 
only  when  the  tube  leaves  at  right  angles  to  the  direction  of  flow 
and  when  its  orifice  is  flush  with  the  walls  of  the  pipe.     No  tube 
projecting  within  the  pipe  will  give  a  true  pressure  reading,  even 
though  it  be  normal  to  the  direction  of  flow.1 

98.  Measurement  of  Water. — The  chief  difficulty  in  turbine 
testing  is  the  measurement  of  the  water  used.     In  some  commer- 
cial plants  the  circumstances  are  such  that  it  is  scarcely  possible 
to  measure  the  water  at  all  and  in  others  the  expense  is  prohibi- 
tive.    The  necessity  of  cheap  and  accurate  means  of  determining 
the  amount  of  water  discharged  is  imperative. 

The  standard  method  of  measurement  is^by  means  of  a  weir. 
For  large  discharges,  however,  the  expense  of  constructing  a 
suitable  weir  channel  may  be  excessive,  and,  in  case  the  turbine 
discharges  directly  into  a  river,  it  may  be  almost  impossible  to 
construct  it.  In  the  case  of  a  turbine  operating  under  a  low  head 
the  increase  in  the  tail-water  level  caused  by  the  weir  may  cause 
a  serious  decrease  in  head  below  that  normally  obtained.  This 
would  make  the  test  of  little  value.  However,  where  it  is  feasible, 
the  use  of  a  weir  is  a  very  satisfactory  method  and  should  be  pro- 
vided for  when  the  plant  is  constructed.  It  should  be  remem- 
bered, though,  that  all  weir  formulas  and  coefficients  are  purely 

1  Hughes  and  Safford,  "  Hydraulics,"  p.  104. 


144  HYDRAULIC  TURBINES 

empirical  in  their  nature  and  that  the  discharge  as  determined  by 
them  may  be  as  much  as  5  per  cent,  in  error,  unless  standard 
proportions  are  carefully  adhered  to.1 

In  order  to  avoid  the  increase  in  the  tail-water  level  the  use  of 
submerged  orifices  may  be  desirable  in  low-head  plants.  A  sub- 
merged orifice  will  produce  a  certain  elevation  of  the  tail-water 
level,  but  it  will  not  be  as  great  as  the  weir.  At  present  enough 
experimental  data  had  not  been  gathered  to  make  this  method 
applicable  in  general,  but  perhaps  in  the  future  it  may  be  used 
with  fair  success. 

Either  in  the  tail  race  or  in  the  head  race  a  Pitot  tube,  current 
meter  j  or  floats  may  be  used.  These  methods  involve  no  dis- 
turbance of  the  head  under  which  the  turbine  ordinarily  operates, 
but  they  do  require  a  suitable  channel  in  which  the  observations 
can  be  taken.  These  instruments  should  be  in  the  hands  of  a 
skilled  observer  who  understands  the  sources  of  error  attendant 
upon  their  use.2 

The  Pitot  tube  consists  of  a  tube  with  an  orifice  facing  the 
current.  The  impact  of  the  stream  against  this  orifice  produces 
a  certain  pressure  which  is  proportional  to  the  square  of  the 
velocity.  If  h  is  this  reading  in  feet  of  water  and  K  and  experi- 
mental constant,  then 

V  = 


Since  it  would  be  very  difficult  to  determine  accurately  the  height 
of  the  column  of  water  in  a  tube  above  the  level  of  the  stream 
it  is  customary  to  use  two  tubes  and  read  the  difference  between 
the  two.  For  convenience  in  reading,  the  instrument  is  made  so 
that  valves  may  be  closed  and  the  device  lifted  out  of  the  water- 
without  changing  the  levels  of  the  columns,  or  sometimes  both 
columns  may  be  drawn  up  to  a  convenient  place.  The  orifice 
for  this  second  tube  is  usually  in  a  plane  parallel  to  the  direction 
of  flow  and  will  thus  give  a  lower  reading  than  the  other.  It  does 
not,  however,  give  the  value  of  the  pressure  at  that  point,  as 
stated  in  Art.  97.  For  low  velocities  it  is  desirable  to  magnify 
this  difference  in  the  two  readings  and  for  that  purpose  the  orifice 
of  the  second  tube  may  be  directed  down  stream.  Its  reading 
will  then  be  less  than  for  the  one  parallel  to  the  direction  of 

1  See  "Weir  Experiments,  Coefficients,  and  Formulas,"  by  R.  E.  Horton, 
U.  S.  G.  S.  Water  Supply  and  Irrigation  Paper  No.  150,  Revised,  No.  200. 

2  See  Hoyt  and  Grover,  "River  Discharge." 


TURBINE  TESTING  145 

flow.  Such  an  instrument  is  called  a  pitometer,  and  the  value 
of  K  for  it  is  always  less  than  1 .0. 

The  current  meter  is  an  instrument  having  a  little  wheel  which 
is  rotated  by  the  action  of  the  current,  the  speed  of  rotation 
being  proportional  to  the  velocity  of  flow. 

The  Pitot  tube  may  also  be  used  in  a  pressure  pipe.  Since  the 
reading  of  the  impact  tube  alone  will  be  the  sum  of  the  pressure 
head  plus  the  velocity  head  it  will  be  necessary  to  use  two  tubes 
in  the  same  manner  as  in  the  case  of  the  open  channel.  The 
value  of  h  will  be  the  difference  between  these  two  readings, 
and  the  value  of  K  must  be  determined  experimentally.  If, 
however,  only  one  orifice  is  used  and  the  pressure  is  determined 
by  a  piezometer  tube  with  its  orifice  lying  flush  with  the  walls  of 
the  pipe  the  difference  between  these  two  readings  may  be  con- 
sidered equal  to  the  velocity  head,  that  means  the  value  of  K  =  1.0. 

For  the  tangential  water  wheel  the  Pitot  tube  may  also  be 
used  to  determine  the  jet  velocity.  In  such  a  case  only  the 
impact  tube  is  required.  While  it  is  well  to  determine  the  value 
of  K  experimentally,  yet  if  the  tube  is  properly  constructed  it 
may  be  taken  as  1 .0.  A  check  on  this  may  be  obtained  as  follows : 
It  is  probably  true  that  the  maximum  velocity  obtained  at  any 
point  in  the  jet  is  the  ideal  velocity.  The  latter  can  be  computed 
from  the  head  back  of  the  nozzle  and  the  value  of  K  should  be 
such  as  to  make  the  two  agree.  Either  in  the  pipe  or  the  jet  it  is 
desirable  to  take  a  velocity  traverse  across  each  of  two  diameters 
at  right  angles  to  each  other.  In  computing  the  average  velocity 
it  is  necessary  to  weight  each  of  these  readings  in  proportion 
to  the  area  affected  by  them.1 

Chemical  methods  are  often  of  value.  If  the  pipe  line  is 
sufficiently  long  a  highly  colored  stain  may  be  added  to  the  water 
at  intake  and  the  time  noted  that  it  takes  the  color  to  appear  in 
the  tail  race.  From  this  and  the  pipe  dimensions  the  rate  of 
discharge  can  be  computed.  A  second  chemical  method  is 
to  inject  a  salt  solution  into  the  water  of  known  concentration  and 

3  See  "Application  of  Pitot  Tube  to  Testing  of  Impulse  Water  Wheels," 
by  Prof.  W.  R.  Eckart,  Jr.,  Institution  of  Mechanical  Engineers,  Jan.  7,  1910. 
Also  printed  in  Engineering  (London),  Jan.  14,  21,  1910. 

Engineering  News,  Vol.  LIV,  Dec.  21,  1905,  p.  660.     See  also  Zeitschrift  des 
ver.  deut.  Ing.,  Mar.  22,  29,  and  Apr.  5,  1913.     For  useful  information  re- 
garding all  the  methods  of  measurement  given  here  see  Hughes  and  Safford, 
Hydraulics. 
10 


146  HYDRAULIC  TURBINES 

at  a  known  constant  rate.  Samples  of  the  water,  after  thorough 
mixing,  are  taken  and  analyzed.  Knowing  the  amount  of 
dilution  it  is  then  possible  to  compute  the  rate  of  discharge.1 

99.  Measurement  of  Output. — The  determination  of  the  power 
output  of  a  turbine  is  also  a  matter  of  some  difficulty.  Perhaps 
the  most  satisfactory  method  is  to  use  some  form  of  a  Prony 
brake  or  absorption  dynamometer.2 

The  use  of  a  simple  brake  is  restricted  to  comparatively  small 
powers.  For  large  powers  it  becomes  rather  expensive  and  diffi- 
cult. A  good  absorption  dynamometer  may  be  used  satisfac- 
torily for  fairly  large  powers  but  the  drawback  is  one  of  initial 
expense.  In  many  cases  also  where  turbines  are  direct  connected 
to  electric  generators  it  may  be  impossible  to  attach  a  brake  of 
any  kind. 

In  such  cases  it  is  necessary  to  supply  an  electrical  load  for 
the  generator  and  determine  the  generator  efficiency.  However, 
this  method  of  testing  involves  a.  number  of  instrument  readings 
which  may  be  more  or  less  in  error  and  a  rather  complicated 
process  of  computation.  Nevertheless  it  can  be  done  with  very 
satisfactory  results.  One  drawback  about  it  is  that  the  speed 
cannot  be  varied  through  the  same  range  of  values  as  in  the  brake 
test.  The  output  of  the  generator  may  be  absorbed  by  a  water 
rheostat  which  will  furnish  an  absolutely  constant  load.  If  it 
is  a  direct-current  machine  this  rheostat  may  simply  consist  of 
a  number  of  feet  of  iron  wire  wound  on  a  frame  and  immersed  in 
water  to  keep  it  cool.  This  water  should  be  running  water  or  a 
large  pond  so  that  its  temperature  may  not  change.  The  current 
is  shorted  through  this  coil;  the  load  is  varied  by  changing  the 
length  of  wire  in  use.  For  a  three-phase  alternator  the  rheostat 
may  consist  of  three  iron  pipes  at  the  vertices  of  an  equilateral 
triangle  with  a  terminal  connected  to  each.  The  load  is  varied 
by  changing  the  depth  of  immersion  of  the  pipes  in  water.3 

A  good  method  recently  employed  in  a  hydro-electric  plant 
where  there  are  two  or  more  similar  units  is  to  let  one  alternator 
drive  the  other  as  a  synchronous  motor.  The  second  rotates 
the  impulse  wheel  or  reaction  turbine  in  the  reverse  direction. 

1  B.  F.  Groat,  "  Chemi-hydrometry  and  precise  Turbine  Testing,"  Trans. 
A.  S.  C.  E.,  Vol.  LXXX,  p.  951  (1915). 

2  C.  M.  Allen,  "  Testing  of  Water  Wheels  after  Installation,"  Journal 
A.S.  M.E.,  April,  1910. 

•Power,  Vol.  XXXVII,  June  17,  1913,  p.  857. 


TURBINE  TESTING  147 

By  running  varying  quantities  of  water  through  the  latter  it 
is  possible  to  supply  any  constant  load  desired. 

100.  Working  Up  Results. — In  figuring  up  the  results  of  test 
data  it  should  be  borne  in  mind  that  any  single  reading  may  be 
in  error  but  that  all  of  them  should  follow  a  definite  law.     Thus 
a  smooth  curve  should  be  drawn  in  all  cases.     Also  if  any  readings 
should  follow  a  law  which  is  any  approach  to  a  straight  line  it  is 
better  to  work  from  values  given  by  that  line  rather  than  from 
the  experimental  values  themselves.     Thus  if  a  turbine  be  tested 
at  constant  gate  opening  and  at  all  speeds,  the  curve  showing  the 
relation  between  speed  and  efficiency  may  be  drawn  at  once  from 
the  experimenal  data.     However,  a  more  accurate  curve  can  be 
constructed  by  noting  that  the  relation  between  speed  and  brake 
reading  is  a  fairly  straight  line.     See  Fig.  86.     TlnV  is  not  a 
straight  line  but  the  curvature  is  not  very  marked  so  that  it  may 
be  drawn  readily  and  accurately.     Values  given  by  this  curve 
may  then  be  used  for  constructing  the  efficiency  curve.     Again, 
when  a  turbine  is  tested  at  constant  speed,  it  should  be  noted 
that  the  relation  between  input  and  output  is  not  a  straight  line 
absolutely,  but  it  is  approximately  so.     If  any  point  falls  de- 
cidedly off  from  a  straight  line  it  is  probably  in  error.     From 
the  line  giving  the  relation  between  input  and  output  the  effi- 
ciency curve  may  be  constructed. 

In  computing  the  true  power  in  a  jet  it  might  also  be  noted  that 
it  is  not  that  given  by  using  the  square  of  the  average  velocity  but 
something  1  or  2  per  cent,  higher  than  that.  The  reason  is  that 
the  velocity  throughout  the  jet  varies  and  the  summation  of  the 
kinetic  energy  of  all  the  particles  is  not  that  obtained  by  using  the 
average  velocity.1 

101.  Determination  of  Mechanical  Losses. — With  the  tangen- 
tial water  wheel  the  mechanical  losses  will  consist  of  bearing 
friction   and   windage.     With   the   reaction   turbine   they   will 
consist  of  the  bearing  friction  and  the  disk  friction  due  to  the 
drag  of  the  runner  through  the  water  in  the  clearance  spaces. 
There  are  several  ways  of  determining  this  but  the  retardation 
method   is   probably  as  satisfactory  as  any.     The  turbine  is 
brought  up  to  as  high  a  speed  as  is  possible  or  desirable  and  the 
power  shut  off.     As  the  machine  slows  down  readings  of  instan- 

1  L.  M.  Hoskins,  "Hydraulics,"  p.  119. 

L.  F.  Harza,  Engineering  News,  Vol.  LVII,  Mar.  7,  1907,  p.  272. 
See  also  Prof.  Eckart's  paper  previously  mentioned. 


148 


HYDRAULIC  TURBINES 


taneous  speed  are  taken  every  few  seconds  and  a  curve  plotted 
between  instantaneous  speed  and  time  as  shown  in  Fig.  105. 
Instantaneous  speed  may  be  determined  by  a  tachometer,  by  a 
voltmeter,  or  by  an  ordinary  continuous  revolution  counter. 
With  the  latter  the  total  revolutions  are  read  every  few  seconds 
without  stopping  it;  the  difference  between  two  consecutive 
readings  will  then  enable  us  to  find  the  average  speed  correspond- 
ing to  the  middle  of  this  time  interval. 

The  power  lost  at  any  speed  is  equal  to  a  constant  times  the 
subnormal  to  the  curve  at  that  speed.  If  L  equals  the  power  lost 
then 

L  =  K  X  BD. 

For  the  proof  of  this  proposition  see  Appendix  A,  II. 


B  D  B  D 

Time 
FIG.  105. — Retardation  curves. 

To  determine  the  value  of  the  constant  K  a  second  run  is 
necessary  with  a  definite  added  load.  This  load,  which  may  be 
small,  may  be  obtained  by  closing  the  armature  circuit  on  a 
resistance  if  a  generator  is  used  in  the  test  or  by  applying  a 
known  torque  if  a  Prony  brake  is  used.  With  the  first  method 
a  watt-meter  should  be  used  and  the  load  kept  constant  for  a 
limited  range  of  speed,  with  the  second  method  the  torque  should 
be  kept  constant  for  a  limited  range  of  speed.  If  this  known 
added  load  be  denoted  by  M  we  then  have 

L  +  M  =  K  X  B'D'. 

Since  L  is  the  only  unknown  quantity  except  K  it  may  be  elimi- 
nated from  these  two  equations  and  we  have 


K  = 


M 


B'D'  -  BD 


TURBINE  TESTING  149 

In  some  work  that  the  author  has  done  this  method  has  proven 
to  be  very  reliable  and  has  checked  with  values  of  friction  and 
windage  losses  as  determined  by  other  methods. 

102.  QUESTIONS  AND  PROBLEMS 

1.  What  is  the  value  of  testing  a  turbine  upon  installation?     What  is  the 
value  of  testing  one  that  has  been  in  operation  for  some  time?     What  is 
the  value  of  a  Holyoke  test  to  the  purchaser  of  another  runner  but  of  similar 
type?     Is  it  cheaper  to  increase  the  power  output  of  a  plant  by  additional 
construction  or  by  improving  the  efficiency? 

2.  What  are  the  various  purposes  for  which  turbine  tests  may  be  con- 
ducted?    What  conditions  would  be  varied  for  each  of  these  and  what  kept 
constant? 

3.  What  methods  of  measuring  the  rate  of  discharge  are  usually  employed 
in  turbine  tests? 

4.  In  what  ways  may  the  power  output  of  a  waterwheel  be  absorbed  and 
measured? 

5.  A  case  is  reported  where  tests  conducted  at  an  expense  of  $5000  resulted 
in  changes  which  improved  the  efficiency  of  the  turbines  1  per  cent.     If  the 
capacity  of  the  plant  is  100,000  h.p.  and  1  h.p.  is  worth  $100,  what  would 
be  the  value  of  the  gain  in  efficiency,  assuming  the  changes  cost  $20,000? 

6.  In  the  Cedars  Rapids  turbines  the  area  of  the  water  passages  at  en- 
trance to  the  casing  =  1080  sq.  ft.  per  unit,  elevation  of  section  above  tail 
water  =  10  ft.,  and  pressure  head  at  this  point  =  20  ft.     The  area  of  the 
mouth  of  the  draft  tube  =  1050  sq.  ft.     The  test  showed  the  power  output 
to  be  10,800  h.p.  with  a  rate  of  discharge  of  3450  cu.  ft.  per  second.     Cal- 
culate two  values  of  the  efficiency,  using  two  values  of  the  head. 

7.  In  the  test  of  a  reaction  turbine  the  water  flowing  over  the  weir  in  the 
tail  race  was  found  to  be  39.8  cu.  ft.  per  second.     The  leakage  into  the  tail 
race  was  found  to  be  1  cu.  ft.  per  second.     The  elevation  of  the  center  line 
of  the  shaft  above  the  surface  of  the  tail  water  was  12.67  ft.     The  diameter 
of  the  turbine  intake  was  30  in.  and  the  pressure  at  this  section  was  meas- 
ured by  a  mercury  U  tube.     The  readings  in  the  two  sides  of  the  mercury 
U  tube  were  10.556  ft.  and  0.900  ft.,  the  zero  of  the  scale  being  at  a  level 
3.82  ft.  below  that  of  the  center  line  of  the  turbine  shaft.     The  generator 
output  was  391.8  kw.,  friction  and  windage  13.8  kw.,  iron  loss  2.0  kw.,  and 
armature  loss  4.4  kw.     The  specific  gravity  of  the  mercury  used  was  13.57. 
Find:  Input  to  turbine,  output  of  turbine  (generator  being  excited  from 
another  unit),  efficiency  of  turbine,  efficiency  of  generator,  efficiency  of  set. 

Ans.     h  =  141.80  ft.,  625  w.h.p.,  550  b.h.p.,  0.880,  0.951,  0.837. 


CHAPTER  XI 
GENERAL  LAWS  AND  CONSTANTS 

103.  Head. — The  theory  that  has  been  presented  has  made  it 
clear  that  the  speed  and  power  of  any  turbine  depends  upon  the 
head  under  which  it  is  operated.  The  peripheral  speed  of  any 
runner  may  be  expressed  as  HI  =  $\/2gh.  It  has  also  been 
shown  that  for  the  best  efficiency  <j>  must  have  a  certain  value  de- 
pending upon  the  design  of  the  turbine.  It  is  thus  apparent  that 
the  best  speed  of  a  given  turbine  varies  as  the  square  root  of  the 
head. 

The  discharge  through  any  orifice  varies  as  the  square  root  of 
the  head,  and  a  turbine  is  only  a  special  form  of  discharge  orifice. 
Since  Vi  =  c\/2gh,  and  since  a  definite  value  of  c  goes  with  the 
best  value  of  </>  as  given  above,  it  follows  that  the  rate  of  discharge 
of  a  given  turbine  varies  as  the  square  root  of  the  head. 

Since  the  energy  of  each  unit  volume  of  water  varies  as  the 
head,  and  since  the  amount  of  water  discharged  per  unit  time 
varies  as  the  square  root  of  the  head  it  must  then  be  true  that  the 
power  input  varies  as  the  three  halves  power  of  the  head. 

In  reality  the  rate  of  discharge  through  any  orifice  is  not  strictly 
proportional  to  the  square  root  of  the  head,  that  is,  the  coeffi- 
cient of  discharge  is  not  strictly  a  constant  but  varies  slightly 
with  the  head.  However,  the  variation  in  the  coefficient  is  small 
and  inappreciable  except  for  very  large  differences  in  the  head. 
Therefore  the  above  statements  are  accurate  enough  for  most 
practical  purposes. 

The  theory  has  also  shown  that  the  losses  of  head  in  any  tur- 
bine vary  as  the  squares  of  the  various  velocities  concerned. 
This  rests  upon  the  assumption  that  the  coefficient  of  loss  k  is 
constant  for  all  values  of  h  as  long  as  <f>  remains  constant.  That 
is  probably  not  true,  but  may  be  assumed  as  true  for  all  practical 
purposes.  Since  these  velocities  vary  as  the  square  root  of  the 
head  their  squares  will  vary  as  the  first  power  of  the  head.  The 
amount  of  water  varies  as  the  square  root  of  the  head  and,  since 
the  power  Ipst  is  the  product  of  these  two  items,  it  follows  that 

150 


GENERAL  LAWS  AND  CONSTANTS 


151 


the  various  hydraulic  losses  vary  as  the  three-halves  power  of 
the  head.  As  the  hydraulic  losses  vary  in  just  the  same  propor- 
tion as  the  power  input,  the  hydraulic  efficiency  will  be  independ- 
ent of  the  value  of  the  head.  If  the  mechanical  losses  followed 
the  same  law  then  the  gross  efficiency  would  also  remain 
unchanged.  The  mechanical  losses  really  follow  different  laws 
at  different  speeds,  as  can  be  seen  in  Fig.  106.  The  factors  which 
influence  this  are  rather  complicated  and  it  does  not  seem  possi- 
ble to  lay  down  any  rule  to  express  mechanical  losses  as  a  func- 
tion of  the  speed.  It  is  probably  true,  however,  that  these  losses 
increase  faster  than  the  first  power  of  the  speed  but  not  much 
faster  than  the  square  of  the  speed.  Since  the  speed  varies  as 


0.4 


o.s 


0.2 


0.1 


100 


200 


R.P.: 


300 


400 


FIG.  106. — Friction  and  windage  of  a  24"  tangential  water  wheel. 

the  square  root  of  the  head  it  is  seen  that  if  the  friction  losses 
vary  at  the  same  rate  as  the  hydraulic  losses  they  must  increase 
as  the  cube  of  the  speed.  As  they  do  not  do  so,  it  is  apparent 
that  the  gross  efficiency  will  be  higher  the  higher  the  head  under 
which  the  turbine  operates.  The  change  in  the  gross  efficiency 
with  a  change  in  head  is  most  apparent  when  the  latter  is  very 
low.  As  the  head  increases  the  mechanical  losses  become  of 
smaller  percentage  value  and  the  gross  efficiency  tends  to  approach 
the  hydraulic  efficiency,  which  is  constant,  as  a  limit.  Thus 
there  is  little  variation  in  efficiency  unless  the  head  is  very  low. 
The  mechanical  losses  are  really  comparatively  small  being 
only  from  2  to  5  per  cent,  usually  and  thus  the  change  in  gross 
efficiency  cannot  be  very  great.  For  moderate  changes  in  head 


152  HYDRAULIC  TURBINES 

we  may  then  state  that  the  efficiency  of  a  turbine  remains  con- 
stant as  long  as  the  speed  varies  so  as  to  keep  <j>  constant. 
Therefore  the  power  output  of  a  given  turbine  varies  approxi- 
mately as  the  three-halves  power  of  the  head.  ^ 

This  proposition  is  rather  important  because  it  is  often  nec- 
essary to  test  a  turbine  under  a  certain  head  which  is  different 
from  the  head  under  which  it  is  to  be  run.  The  question  may 
then  arise  as  to  how  far  the  test  results  can  be  applied  to  the  new 
head.  As  long  as  the  two  heads  are  not  radically  different  we 
may  state  that  they  will  apply  directly.  If  there  is  a  large  differ- 
ence in  head  we  may  expect  that  the  efficiency  under  the  higher 
head  may  be  one  or  two  or  more  per  cent,  higher.  This  is  borne 
out  by  some  tests  made  by  F.  G.  Switzer  and  the  author  where  the 
head  was  varied  from  30  ft.  to  175  ft.  and  later  from  9  ft.  to  305  ft. 

It  is  customary  to  state  the  performance  of  a  turbine  under 
one  foot  head .  Then  by  means  of  the  above  relations  we  may  easily 
tell  what  it  will  do  under  any  head.  If  the  suffix  (1)  denotes  a 
value  for  one  foot  head  we  may  then  write, 

N  =  NiVh  (44) 

q  =  qiVh  (45) 

h.p.  =  h.p.ih3/2*  (46) 

It  must  be  noted  that  these  simple  laws  of  proportion  may  be 
applied  only  when  the  speed  varies  with  the  head  in  such  a  way 
as  to  keep  <£  constant.  The  value  of  <f>  need  not  necessarily  be 
that  for  the  highest,  efficiency.  But  if  the  speed  does  not  change 
or  if  it  varies  in  some  other  way  so  that  <£  is  different,  the  results 
under  the  new  head  cannot  be  computed  save  by  complex  equa- 
tions, such  as  those  of  Arts.  87  and  88,  or  by  the  use  of  test  curves 
such  as  those  of  Figs.  95  and  96. 

104.  Diameter  of  Runner. — When  a  certain  type  of  runner  has 
been  perfected  a  whole  line  of  stock  runners  of  that  type  may  then 
be  built  with  diameters  ranging  from  10  to  70  in.  or  more.  All 
of  these  runners  will  be  homologous  in  design,  that  is  they  will 
have  the  same  angles  and  the  same  values  of  the  ratios  x,  y, 
and  B/D.  Each  runner  will  simply  be  an  enlargement  or 
reduction  of  another.  They  will  then  have  the  same  character- 
istics, that  is,  the  same  values  of  <j>e  and  ce)  and  will  therefore 
follow  certain  laws  of  proportion. 

*The  easiest  way  to  find  h**  is  to  note  that  h%  =  h\/h.  This  may  be 
found  in  one  setting  of  the  slide  rule. 


GENERAL  LAWS  AND  CONSTANTS  153 

Since  for  a  given  head  HI  will  have  the  same  value  for  all  of 
them,  it  follows  that  for  a  series  of  runners  of  homologous  design 
the  best  r.p.m.  will  be  inversely  proportional  to  the  diameter. 

Since  the  discharge  through  any  runner  is  equal  to  Aic\/2gh, 
and  since  c  will  have  essentially  the  same  value  for  all  the  runners 
of  such  a  series,  the  discharge  will  be  proportional  to  the  area  A  i. 
But  if  the  runners  are  strictly  homologous  the  area  AI  will  be 
proportional  to  the  square  of  the  diameter.  It  will  therefore  be 
true  that  the  discharge  of  any  turbine  of  the  series  will  be  pro- 
portional to  the  square  of  the  diameter. 

Since  the  power  is  directly  related  to  the  discharge  it  also  fol- 
lows that  the  power  of  the  turbine  is  proportional  to  the  square 
of  the  diameter. 

These  relations  are  of  practical  value  because  if  the  speed, 
discharge,  and  power  of  any  runner  is^nown  by  accurate  test, 
predictions  may  then  be  made  regarding  the  performance  of  any 
other  runner  of  the  series.  These  laws  may  not  hold  absolutely 
in  all  cases  because  the  series  may  not  be  strictly  homologous, 
that  is  the  larger  runners  may  differ  slightly  from  the  smaller 
ones.  Also  it  will  no  doubt  be  true  that  the  efficiency  of  the  larger 
runners  will  be  somewhat  higher  than  that  of  the  smaller  ones. 
It  may  also  be  found  that  careful  tests  of  two  runners  made  from 
the  same  patterns  will  not  give  exactly  the  same  results  due  to 
difference  in  finish  or  other  imperceptible  matters.  Despite 
these  factors,  however,  the  relations  stated  are  true  enough  to  be 
used  for  most  purposes. 

105.  Commercial  Constants.  —  For  a  given  turbine  the  maxi- 
mum efficiency  will  be  obtained  only  for  a  certain  value  of  <t>. 
All  tables  in  catalogs  of  manufacturers  as  well  as  all  values  given 
in  this  chapter  are  based  upon  the  assumption  that  the  speed  will 
be  such  as  to  secure  this  value  of  4>.  Substituting  values  of  N 
and  D  for  HI  in  the  expression  HI  =  <i>\/2gh,  we  obtain 

N  =  184Q.frVfe  (47) 

where  D  is  the  diameter  of  the  runner  in  inches.     From  this  may 
also  be  written 

DN 

</>  =  0.000543  —7=  (48) 


Since  <t>e  is  constant  for  any  series  of  runners  of  homologous 

DN 

design,  it  follows  from  (48)  that  the  expression  —  -r=-  must  remain 


154  HYDRAULIC  TURBINES 

a  constant.     If,  then,  the  best  r.p.m.  of  any  diameter  of  runner 
under  any  head  is  determined,  the  proper  r.p.m.  of  any  other 
runner  of  the  series  under  any  head  may  be  readily  computed. 
For  the  tangential  water  wheel: 

4>.  =  0.43  to  0.47 

DN 

-77  =  790  to  870. 
Vh 

For  the  reaction  turbine  : 

4>e  =  0.55  to  0.90. 


=  =  1050  to  1600. 

Vh 

If  values  outside  these  limits  are  met  with  it  is  because  the  speed 
is  not  the  best  or  because  the  nominal  value  of  D  is  not  the  true 
value. 

106.  Diameter  and  Discharge.  Since,  for  any  fixed  gate  open- 
ing and  a  constant  value  of  <£,  the  rate  of  discharge  of  any  runner 
is  proportional  to  the  square  of  its  diameter  and  to  the  square 
root  of  the  head,  we  may  write 

q  =  KiD*Vh  (49) 

The  value  of  K\  depends  upon  the  velocity  Vi  and  the  area  A\. 
The  former  depends  upon  the  value  of  c  (Art.  83),  and  the  latter 
depends  upon  the  diameter  D,  the  height  of  the  runner  B  (Fig. 
34),  the  value  of  the  angle  «i,  and  also  the  number  of  buckets 
and  guides. 

Since  there  are  so  many  factors  involved,  it  will  be  seen  that  a 
given  value  of  KI  can  be  obtained  in  several  ways.  For  some 
purposes  it  might  be  convenient  to  express  these  items  by  sepa- 
rate constants  but  for  the  present  purpose  it  will  be  sufficient 
to  cover  all  of  them  by  the  one  constant. 

The  lowest  value  of  KI  will  be  obtained  for  the  tangential  water 
wheel  with  a  single  jet.  For  this  type  of  wheel  there  is  evidently 
no  minimum  value  of  KI  below  which  we  could  not  go.  The 
maximum  value  of  KI  is,  however,  fixed  by  the  maximum  size  of 
jet  that  may  be  used.  (See  Art.  30  and  Art.  74.)  Using  this 
maximum  size  of  jet  we  obtain  a  value  of  KI  =  0.0005.  How- 
ever the  more  usual  value  is  about  KI  =  0.0003.  There  is  seldom 
any  reason  for  using  a  large  diameter  of  wheel  with  a  small  jet 
and  so  much  lower  values  are  rare. 


GENERAL  LAWS  AND  CONSTANTS  155 

With  the  reaction  turbine  the  lowest  values  of  K\  would  be 
obtained  with  type  /  in  Fig.  34  and  the  highest  with  type  IV. 
The  value  of  the  area  AI  is  proportional  to  the  sine  of  a\  and  nor- 
mally small  values  of  on  go  with  small  values  of  the  ratio  B/D. 
Taking  the  usual  values  that  go  with  either  extreme  we-  get  a 
minimum  value  of  KI  =  0.0010  and  a  maximum  value  KI  = 
0.050.  These  are  not  absolute  limits  but  they  cannot  be  ex- 
ceeded very  much  and  to  do  so  at  all  would  mean  to  extend  our 
proportions  of  design  beyond  present  practice.  For  the  usual 
run  of  stock  turbines  values  of  KI  vary  from  0.005  to  0.025.  To 
summarize  : 

For  the  tangential  water  wheel  K±  =  0.0002  to  0.0005 
For  the  reaction  turbine  KI  =  0.001  to  0.050 

107.  Diameter  and  Power.  —  Since  the  power  of  any  runner  is 
proportional  to  the  square  of  Ihe  diameter  and  to  the  three-halves 
power  of  the  head,  we  may  write 


h.p.  =  K2D2h*  (50) 

As  the  power  is  directly  dependent  upon  the  discharge  it  is 
evident  that  the  discussion  in  the  preceding  article  will  apply 
equally  well  here.  K2  may  be  computed  directly  from  KI  if  the 
efficiency  is  known,  or  it  may  be  determined  independently  by 
test. 

For  the  tangential  water  wheel  K2  =  0.000018  to  0.000045 
For  the  reaction  turbine  K2  =  0.00008  to  0.00450 

108.  Specific  Speed.  —  In  Art.  105  we  have  the  relation  between 
diameter  and  r.p.m.;  in  Art.  107  we  have  the  relation  between 
diameter  and  power.     It  is  now  desirable  to  establish  the  rela- 
tion between  r.p.m.  and  power  as  follows: 
From  (47) 


N 
From  (50) 


Substituting  the  above  value  of  D  in  the  second  expression  we 
have 

/—  18400  V/i 

VK*  — 


156  HYDRAULIC  TURBINES 

Letting  N8  stand  for  the  constant  factors  and  rearranging  we 
have 


This  expression  is  a  very  useful  factor  and  is  called  the  specific 
speed.  It  is  also  called  unit  speed  or  type  characteristic  or  char- 
acteristic speed  by  various  writers.  Its  physical  meaning  can  be 
seen  as  follows:  If  the  head  be  reduced  to  1  ft.  then  Ns  =  N  \/Ti~p. 
By  then  varying  the  diameter  of  the  runner  the  value  of  N 
will  change  in  an  inverse  ratio,  but  the  square  root  of  the  horse- 
power varies  directly  as  D.  Thus  the  product  of  the  two  or  Ns 
remains  constant  for  all  values  of  D  as  long  as  the  series  is  homo- 
logous. If  a  value  of  D  be  chosen  which  will  make  the  h.p.  = 
1.0  when  h  =  1  ft.,  we  then  have  N8  =  N. 

That  is,  the  specific  speed  is  the  speed  at  which  a  turbine 
would  run  under  one  foot  head  if  its  diameter  were  such  that  it 
would  develop  1  h.p.  under  that  head.  The  specific  speed  is 
also  an  excellent  index  of  the  class  to  which  a  turbine  belongs  and 
hence  the  term  type  characteristic  is  very  appropriate.  There 
is  no  standard  symbol  used  by  all  to  denote  this  constant  though 
Ns  is  quite  common.  Other  notations  are  Nu,  KT,  and  numerous 
others.  In  Europe  the  specific  speed  will  be  expressed  in  metric 
units;  to  convert  from  one  to  the  other  multiply  N,  in  English 
units  by  4.45. 

It  should  be  noted  that  the  power  to  be  used  in  this  formula  is 
the  power  output  of  the  machine.  Thus  the  efficiency  is  involved 
in  the  value  of  Ns,  though  it  does  not  appear  directly.  In  the 
case  of  a  Pelton  wheel  with  two  or  more  nozzles,  the  power  to  be 
used  is  that  corresponding  to  only  one  jet.  In  the  case  of  multi- 
runner  units,  the  specific  speed  should  be  computed  for  the 
power  of  one  runner. 

For  any  turbine  the  value  of  N8  is  a  constant,  so  long  as  the 
speed  of  the  turbine  is  varied  as  the  square  root  of  the  head. 
For  if  N  varies  as  \/h  and  the  power  varies  as  h?*,  it  is  seen  that 
N\/h.p.  varies  as  hy*.  Also  for  a  series  of  homologous  runners  the 
square  root  of  the  power  increases  with  D  directly  while  the  speed 
N  varies  inversely.  Thus  the  factor  is  a  constant  for  all  turbines 
of  the  same  type. 

The  value  of  the  specific  speed  is  ordinarily  computed  by  equa- 


GENERAL  LAWS  AND  CONSTANTS  157 

tion  (51),  since  this  involves  the  quantities  with  which  the 
engineer  is  most  concerned.  But  the  great  practical  value  of  this 
factor  in  turbine  work  is  such  as  to  make  it  worth  while  to  derive 
this  expression  in  other  ways  and  in  terms  of  other  quantities. 

Thus 

M,  =  irDN/720  =  <t>\/2gh 

From  which  D  =  72Q<f>\/2gh/TrN  (52) 

Also,  if  B  =  mD, 

q  =  (0.957rBD/144)yrl  =  0.95irroZ>V\/ty£/144  (53) 

where  0.95  is  a  factor  to  compensate  for  the  area  taken  up  by  the 
runner  vanes. 
Since          B.h.p.  =  wqhe/550 

B.h.p.  =  0.95  wir^/Zgm  D2cr.h-  e/144  X  550  (54) 

Eliminating  D  between  the  simultaneous  equations  (52)  and  (54) 
and  reducing,  we  have  (giving  </>  the  special  value  <f>e) 


N,  =  -..  _  252*.v%XwX.«  (55) 

This  equation  shows  how  the  value  of  the  specific  speed  may  be 
varied  in  the  design  by  means  of  the  factors  <£e,  cr,  and  m.3 
An  instructive  form,  however,  is  that  of  Lewis  F.  Moody,  in 
which  the  diameter  of  the  draft  tube  is  represented  as  nD,  and 
the  discharge  velocity  head  V22/2g  =  Lh,  where  L  is  the  frac- 
tional part  of  the  head  h  that  is  lost  at  discharge  from  the  runner. 
(Of  course  an  efficient  draft  tube  is  relied  upon  to  recover  a  part 
of  this).  With  these  we  may  write 

'  (56) 


Substituting  this  expression  for  q  in  that  for  horsepower,   we 
obtain 

B.h.p.  =  ww\/2gn2D2\/Lhe/4:  X  144  X  550  (57) 

Eliminating  D  between  the  simultaneous  equations  (52)  and  (57) 
and  reducing,  we  have 


N.  =  -  -  129.5n*.VvEv7  (58) 

/l/4 

xln  a  similar  manner  the  specific  speed  for  a  Pelton  wheel  may  be  shown 
to  be,  N»  =  1290e\/clle^  where  d  =  jet  diameter  in  inches.  Since,  for  the 
impulse  wheel  <f>e  and  cw  are  practically  constant  this  may  be  reduced  to 
N.  =  53.7  i 


158  HYDRAULIC  TURBINES 

While  the  equation  in  this  form  shows  how  the  specific  speed  may 
be  varied  in  design  by  changing  the  factors  n,  <j>e,  and  L,  its  chief 
use  is  in  showing  the  limit  which  the  specific  speed  approaches. 
Thus  to  increase  the  value  of  Na,  the  ratio  n  may  be  increased. 
But  it  will  soon  reach  a  definite  limit.  The  factor  <j>e  may  also  be 
increased,  but  it  also  will  reach  a  definite  limit,  which  is  something 
under  1.0.  The  efficiency  cannot  readily  be  increased  any  more 
than  for  lower  specific  speed  runners  and  as  a  matter  of  fact,  is 
already  decreasing.  Thus  after  these  factors  have  reached  their 
maximum  limits,  so  that  they  may  be  assumed  to  be  constant,  the 
only  means  of  increasing  Na  any  further  would  appear  to  be  by 
increasing  L.  Thus 

Ns  cc  vVl  or  L  oc  N44  (59) 

But  after  this  limit  is  passed  so  that  equation  (59)  applies, 
the  outflow  loss  increases  much  faster  than  the  specific  speed. 
Even  with  the  best  of  draft  tubes  a  certain  percentage  of  L 
must  be  lost  eventually  and  hence  e  is  rapidly  reduced.  The 
outflow  conditions  thus  impose  a  maximum  limit  upon  N,. 

For  the  lower  values  of  Ns  the  outflow  loss  becomes  of  small 
consequence,  but  other  factors  then  enter.  The  chief  of  these  are 
the  leakage  losses  and  the  disk  friction.  For  with  small  values 
of  the  specific  speed  the  runner  becomes  relatively  large  in 
diameter  and  correspondingly  narrow.  The  area  of  the  spaces 
through  which  water  can  leak  becomes  of  greater  percentage  as 
compared  with  the  area  through  the  runner.  And  the  percentage 
of  the  power  consumed  in  rotating  the  large  diameter  runner 
through  the  water  in  the  clearance  spaces  becomes  of  increasing 
importance.  If  we  assume  that  the  power  lost  in  disk  friction 
varies  as  DbN3,  it  may  be  readily  shown  by  combining  this  with 
equations  (51)  and  (52)  that  the  power  so  lost  varies  as  <l>b./N82. 
After  <t>e  has  been  reduced  to  its  minimum,  which  approaches 
0.50  as  a  limit,  any  further  decrease  in  N,  increases  the  disk  fric- 
tion loss  much  more  rapidly.  Also  as  4>e  is  decreased  ce  must  in- 
crease (approaching  unity  as  a  limit),  as  shown  by  equation 
(39),  and  consequently  pi  decreases  (approaching  zero  as  a  limit). 
But  this  is  undesirable,  due  to  the  danger  of  oxidation  of  parts  of 
the  runner. 

In  view  of  these  facts,  it  may  be  shown  that  the  minimum  allow- 
able value  for  the  specific  speed  of  a  reaction  turbine  is  about  10, 


GENERAL  LAWS  AND  CONSTANTS*  159 

though  there  are  a  few  extreme  cases  of  design  that  have  'carried 
it  as  low  as  about  9.  With  present  draft  tube  construction,  the 
maximum  limit  for  the  specific  speed  of  a  reaction  turbine  is 
about  100,  though  values  as  high  as  130  are  attainable  at  some 
sacrifice  of  efficiency.  The  usual  range  in  practice  varies  from 
about  20  to  80. 

A  very  recent  type  of  turbine  runner  proposed  by  Nagler  is 
of  an  axial  flow  type  and  is  similar  to  a  screw  propeller.  The 
present  specific  speed  of  this  type  is  165  and  it  is  possible  that 
this  may  be  extended  in  the  future.1 

The  impulse  turbine  runs  in  air  and  thus  the  disk  friction  loss 
for  it  becomes  windage  loss,  which  is  of  less  consequence.  There 
can  be  no  leakage  loss  with  this  type  and  also  the  reduction 
of  the  pressure  to  atmospheric  gives  rise  to  no  trouble.  Hence 
this  type  of  turbine  is  suitable  for  specific  speeds  below  those  for 
the  reaction  turbine.  For  the  tangential  water  wheel  there  is 
no  definite  lower  limit  to  its  specific  speed,  save  that  the  windage 
loss  affects  it  in  a  similar  manner  to  the  disk  friction  in  the  case 
of  the  low-speed  reaction  turbine.  But  as  the  specific  speed  of  a 
Pelton  wheel  is  increased  the  size  of  the  jet  must  become  larger 
in  proportion  to  that  of  the  wheel  and  for  the  reasons  already 
given  there  is  a  limit  to  this.  The  further  increase  in  ratio  of 
jet  diameter  to  wheel  diameter  causes  the  efficiency  to  rapidly 
decrease,  due  to  loss  of  water  past  the  buckets.  There  have  been 
cases  of  tangential  wheels  with  specific  speeds  of  less  than  1  and 
maximum  values  of  6,  though  the  latter  involves  some  sacrifice 
of  efficiency.  The  usual  range  in  practice  is  from  3  to  4.5. 

It  will  be  seen  that  there  is  a  gap  in  the  values  of  N8  between 
the  tangential  water  wheel  and  the  reaction  turbine.  Similar 
gaps  are  also  found  for  the  values  of  <£e,  KI,  and  K2.  In'Europe  a 
few  two-stage  radial  inward-flow  reaction  turbines  have  been 
built  and  these  could  have  lower  values  of  the  specific  speed  than 
10.  And  by  the  use  of  two  or  more  nozzles  on  one  impulse  wheel 
runner,  the  value  of  N8  for  the  tangential  wheel  can  be  increased 
above  the  5  or  6  set  'as  the  limit  for  the  single  nozzle.  Thus  the 
entire  field  can  be  covered. 

To  recapitulate: 

For  the  tangential  water  wheel    N8  =  3.5  to  4.5  (6  max.) 
For  the  reaction  turbine  N8  =  10  to  100. 

Jour,  of  the  Amer.  Soc.  of  Mech.  Eng.,  Dec.,  1919. 


160 


HYDRAULIC  TURBINES 


109.  Determination  of   Constants. — The  constants  given  in 
this  chapter  may  be  computed  from  theory,  but  for  practical 
use  should  be  secured  from  test  data.     The  catalogs  of  turbine 
manufacturers    usually    contain    tables    giving   the    discharge, 
power,  and  speed  of  different  diameters  of  runners  under  various 
heads.     As  these  tables  are  supposed  to  be  based  upon  tests 
they  may  be  used  for  the  determination  of  these  factors.     If 
all  the  runners  of  the  series  were  strictly  homologous  it  would 
be  necessary  to  compute  these  constants  for  one  case  only. 
Actually  variations  will  exist  with  different  diameters  of  runners 
and  thus  there  will  be  some  variation  in  the  values  secured. 
Since  each  manufacturer  usually  makes  several  lines  of  runners 
so  as  to  cover  the  field  to  better  advantage,  there  will  be  as  many 
distinct  values  of  these  constants  as  he  makes  types  of  runners. 
If  the  catalog  tables  are  purely  fictitious  then  the  computations 
based  upon  them  will  not  be  very  reliable. 

110.  Illustrative  Case. — In  order  to  illustrate  the  preceding 
article  the  following  tables  are  given.     For  the  sake  of  compari- 
son only  two  firms  out  of  many  are  chosen  for  this  case.     The 
values  given  are  based  upon  catalog  tables.     Since  K%  depends 
upon  Ki  it  has  been  omitted  to  save  space. 

TABLE  3. — JAMES  LEPFEL  AND  Co. 


Type 

<i> 

Ki 

N, 

Standard 

0  722-0.727 

0.0061-0.0064 

30.8-32.6 

Special        

0.750-0.779 

0.0094-0.0097 

41.6-43.2 

Samson                 

0.838-0.844 

0.0170-0.0171 

61.5-61.9 

Improved  Samson           .... 

0.856-0  886 

0  0220-0  0220 

71.0-73.5 

TABLE  4. — DAYTON  GLOBE  IRON  WORKS  Co. 


Type 

<t> 

/Ci 

N, 

High  head  type  

0.578-0.585 

0.0051-0.0064 

22.8-26.0 

American            

0.662-0.704 

0.0054-0.0080 

25.0-32.3 

Special  New  American  
Improved  New  American.  .  . 

0.697-0.727 
0.886-0.944 

0.0175-0.0205 
0.0233-0.0263 

50.0-57.4 
78.2-80.5 

This  table  shows  the  variation  in  constants  that  might  be  ex- 
pected, and  shows  also  how  each  firm  attempts  to  cover  the 
ground.  It  will  be  noticed,  however,  that  the  two  do  not  agree 
in  all  respects.  Thus  suppose  a  turbine  was  desired  whose  speci- 


GENERAL  LAWS  AND  CONSTANTS  161 

fie  speed  was  42.  The  "Special"  turbine  of  the  Leffel  Co.  would 
fulfill  the  conditions,  but  the  Dayton  Globe  Iron  Works  Co. 
have  no  line  of  turbines  that  would  exactly  answer  the  require- 
ment. The  latter  firm  might  furnish  a  turbine  that  would  have 
the  required  specific  speed  but  it  would  have  to  be  a  special 
design — it  would  not  be  a  stock  turbine,  and  would  therefore 
be  more  expensive. 

111.  Uses  of  Constants. — After  these  factors  are  determined 
it  will  then  be  easy  to  find  what  results  may  be  secured  for  any 
size  turbine  of  the  same  design  under  any  head.  Another  use 
for  them  is  that  when  the  limits  are  fixed  they  will  enable  one 
to  tell  what  is  possible  and  what  is  not.  In  the  next  chapter 
it  will  be  shown  how  they  are  of  direct  use  also  in  the  selection 
of  a  turbine. 

112.  NUMERICAL  ILLUSTRATIONS 

1.  The  test  of  a  16-in.  runner  under  a  25-ft.  head  gave  the  following  as 
the  best  results:  N  =  400,  q=  17.5  cu.  ft.  per  second,   h.p  =  39.8.     Find 
the  constants. 

From  (48)  </>  =  0.000543  16'X400  =  0.696 

• '  •  1 1  •  o 

From  (49)  Kl  =  ^J^g  =  0.01368 

39  8 
From  (50)  K*  =  lffl  x  12g  =  0.00124 

From  (51)  N.  =  ***£***  =  45'2 

2.  Suppose  that  a  40-in.  runner  of  the  same  design  as  in  problem  (1)  is 
used  under  a  150-ft.  head.     Compute  the  speed,  discharge,  and  horse-power. 

1840  X  0.696  X  12.25 
From  (47)  N  =  -  4Q  -  =  392  r.p.m. 

From  (49)  q  =  0.01368  X  1600  X  12.25  =  268  cu.  ft.  per  second 
From  (34)  0.00124  X  1600  X  1838  =  3650  h.p. 

3.  Suppose  that  turbines  of  the  type  in  problem  (1)  were  satisfactory  for 
a  certain  plant  but  that  the  number  of  the  units  (and  consequently  the 
power  of  each)  and  the  speed  has  not  been  decided  upon.     If  the  head  is 
150ft.,  then  by  (51) 

N  xVh'.p'.  =  45.2  X  525  =  23,730. 

By  the  use  of  different  diameters  of  runners  of  this  one  type  the  following 
results  can  be  secured : 

14, 100  h.p.  at  200  r.p.m. 

6,250  h.p.  at  300  r.p.m. 

3,520  h.p.  at  400  r.p.m. 

2,250  h.p.  at  500  r.p.m. 

11 


162  HYDRAULIC  TURBINES 

1,560  h.p.  at  600  r.p.m. 

,.  1,150  h.p.  at  700  r.p.m. 

878  h.p.  at  800  r.p.m. 
695  h.p.  at  900  r.p.m. 

If  the  capacity  of  the  plant  were  25,000  h.p.  it  might  then  have  4  units  at 
300  r.p.m.,  16  units  at  600  r.p.m.,  or  36  units  at  900  r.p.m.  If  none  of  the 
possible  combinations  were  suitable  it  would  be  necessary  to  use  another 
type  of  turbine — that  is  one  with  a  different  value  of  Ns. 

By  equation  (50)  the  diameters  are  found  to  be  52.3  in.,  26.2  in.,  and 
17.5  in.  for  300,  600,  and  900  r.p.m.  respectively. 

4.  Compute  values  of  <f>,  Ki,  Kz,  and  N8  for  each  of  the  turbines  whose 
tests  are  given  in  Appendix  C:  (a)  for  the  point  of  highest  efficiency,  (6) 
for  the  point  of  maximum  power. 


113.  QUESTIONS  AND  PROBLEMS 

1.  How  do  the  speed,  rate  of  discharge,  power,  and  efficiency  of  a  turbine 
vary  with  the  head,  the  value  of  0  remaining  constant?     Why? 

2 .  Suppose  the  speed  of  a  turbine  remains  constant  while  the  head  changes, 
how  will  the  rate  of  discharge,  power  and  efficiency  vary?     What  is  neces- 
sary in  order  to  answer  this  question? 

3.  How  do  the  speed,  power,  and  efficiency  vary  with  the  diameter  of  a 
series  of  homologous  runners?     Why?     How  do  these  quantities  change 
when  both  the  head  and  diameter  are  different,  the  runners  being  of  the 
same  type,  however? 

4.  What  is  the  physical  meaning  of  the  term  "specific  speed?"     Why  are 
the  terms  "type  characteristic"  and  "characteristic  speed"  also  appropriate? 
How  may  the  value  of  this  factor  be  changed  in  the  design  of  the  runner? 

5.  What  limits,  the  maximum  and  minimum  values  of  the  specific  speed 
for  reaction  turbines?     For  impulse  wheels?     Why  do  the  latter  have 
lower  specific  speeds  than  the  former? 

6.  If  a  turbine  gives  an  efficiency  of  82  per  cent,  when  tested  under  a 
head  of  10  ft.  what  would  you  estimate  its  efficiency  to  be  if  installed  under 
a  head  of  100  ft.?     Under  a  head  of  225  ft.?     If  the  test  of  a  27-in.  runner 
under  a  head  of  150  ft.  gives,  as  the  best  results,  N  =  600,  q  =  40,  h.p.  = 
550,  what  will  be  the  speed,  rate  of  discharge,  and  power  of  a  54-in.  runner 
of  the  same  type  under  a  head  of  50  ft.  ? 

7.  If  a  turbine  is  desired  to  run  at  300  r.p.m.  under  a  head  of  60  ft.,  what 
are  the  minimum  and  maximum  diameters  of  runners  that  might  be  used? 
If  30  cu.  ft.  of  water  per  second  is  to  be  used  under  a  head  of  60  ft.,  what 
range  of  diameters  might  be  employed? 

8.  Suppose  that  a  type  of  turbine,  whose  specific  speed  is  80,  is  suitable 
for  use  in  a  certain  plant  where  the  head  is  16  ft.     What  combinations  of 
h.p.  and  r.p.m.  are  possible? 

9.  If   a  tangential  water  wheel  was  desired  to  deliver  1000  h.p.  under 
150  ft.  head,  what  r.p.m.  could  be  used?     How  high  a  speed  could  be  ob- 
tained with  a  reaction  turbine? 


GENERAL  LAWS  AND  CONSTANTS  163 

10.  Wouldtit  be  possible  to  obtain  a  5000  h.p.  turbine  to  run  at  600  r.p.m. 
under  a  50-ft.  head?     What  could  be  done  to  secure  that  power?       To  secure 
600  r.p.m.? 

11.  A  reaction  turbine  is  designed  so  that  <f>e  =  0.72,  B/D  =  0.29,  cr  = 
0.20,  and  the  efficiency  may  be  assumed  =  0.88.     What  is  the  value  of  the 
specific  speed?     Compute  the  probable  values  of  the  specific  speeds  for  the 
four  types  of  turbines  shown  in  Fig.  34,  making  whatever  assumptions  are 
necessary. 

12.  The  turbine,  whose  dimensions  are  given  in  problem  (11)  had  a  value 
of  n  =  1.04.     Find  the  per  cent,  of  the  total  head  that  is  equal  to  the  velocity 
head  at  discharge  from  the  runner? 


CHAPTER  XII 
TTRBINE  CHARACTERISTICS 

114.  Efficiency  as  a  Function  of  Speed  and  Gate  Opening. 
In  Fig.  87,  page  112,  it  has  been  shown  how  the  power,  and  hence 
the  efficiency,  of  an  impulse  turbine  varies  with  the  speed  for 


Values  of  0 
FIG.  107. — Characteristics  of  high-speed  runner. 


Ng  =  93. 


any  gate  opening;  and  in  Fig.  91,  page  115,  how  the  efficiency 
varies  with  the  power  at  different  gate  openings  at  a  uniform 
speed,  the  head  being  constant  in  both  cases.  Similar  curves 
for  a  reaction  turbine  are  shown  in  Figs.  96  and  98. 

It  should  be  noted  that  the  value  of  <£  for  the  highest  efficiency 

164 


TURBINE  CHARACTERISTICS 


165 


at  one  gate  opening  is  not  the  same  as  that  for  any  other  gate 
opening.  This  is  best  shown  by  Fig.  107,  and  reasons  for  it 
are  given  in  Arts.  76  and  91.  Hence  the  speed  that  is  most  effi- 
cient for  one  gate  opening  is  not  exactly  the  best  for  any  other 
gate  opening.  Also  the  values  of  efficiency  vary  for  the  different 
gate  openings  and  the  maximum  efficiency  will  be  found  at  some- 
thing less  than  "full"  gate.1  Therefore,  in  general,  the  maxi- 
mum efficiency  and  the  maximum  power  are  found  at  different 
gate  openings  and  different  speeds. 


0.8       1.0       1.2       1.4       1.6       L8       2.0       2.2       2.4       2.6       • 
Brake  Horse-Power  under  1  Ft.  Head 
FIG.  108. — Efficiency-power    curves    for    different    speeds    under    same    head. 

Thus  in  Fig.  107  the  maximum  value  of  the  efficiency  is  found 
at  0.820  gate  and  at  such  a  speed  that  <j>e  =  0.780,  but  the  maxi- 
mum power  is  found  at  1.103*  gate  and  at  such  a  speed  that 
0  =  1.03.  The  efficiency  in  the. former  case  is  0.88  and  in  the 
latter  0.77.  In  Fig.  108  are  shown  efficiency  curves  as  a  function 
of  power  for  values  of  0  =  0.64,  0.78,  and  1.03. 

1  This  statement  does  not  hold  in  the  case  of  the  cylinder  gate  turbine, 
where  maximum  power  and  maximum  efficiency  coincide  at  full  gate,  but 
this  type  is  of  little  importance  at  present. 

*  The  numbers  indicating  the  extent  of  the  gate  opening  are  purely 
arbitrary  and  1.0  does  not  necessarily  indicate  the  maximum  gate  opening. 
This  will  be  explained  subsequently. 


166 


HYDRAULIC  TURBINES 


In  selecting  the  proper  speed  for  a  given  turbine  a  number  of 
operating  factors  must  be  taken  into  consideration.  If  the  head 
is  constant  and  the  load  is  constant,  it  may  be  possible  to  operate 
the  turbine  near  the  point  of  maximum  efficiency  most  of  the 
time.  In  this  case  it  might  be  desirable  to  select  the  speed  giv- 
ing the  maximum  efficiency,  or  0  =  0.78  in  the  case  above.  But 
if  the  load  is  variable  and  especially  if  it  is  apt  to  be  light  for  long 
periods  of  time  a  lower  value  of  the  speed  might  give  a  higher 
average  efficiency,  though  the  peak  is  not  so  high.  On  the  other 
hand  it  may  be  deemed  worth  while  to  sacrifice  efficiency  for  the 
sake  of  capacity  and  increased  speed,  which  could  be  attained  by 
using  the  higher  values  of  the  speed.  It  should  be  borne  in  mind 
that  some  of  these  results  might  be  better  attained  with  another 
type  of  turbine,  but  the  latter  is  a  subject  for  consideration  in 


2000       4000        6000         8000       10000      12000     14000      16000     18000     20000 

Brake  Horse  Power 

FIG.  109. — Power  and  efficiency  of  a  turbine  at  constant  speed  under  different 

heads. 

the  next  chapter.  We  are  here  studying  the  possibilities  of 
a  single  turbine  or  at-  least  a  single  type  of  turbine. 
,  It  must  be  remembered,  as  explained  in  Art.  20,  that  the  head 
is  apt  to  vary  for  many  water  power  plants,  especially  those  under 
low  head.  If  the  head  decreases  in  time  of  flood,  the  power 
output  of  the  turbine  may  be  seriously  reduced.  Under  these 
circumstances  the  important  consideration  is  the  maximum 
power  output.  Since  there  is  a  superabundance  of  water  for  the 
time  being,  efficiency  is  a  secondary  consideration.  While 
efficiency  under  the  normal  head  is  of  importance,  it  might  be 
sacrificed  to  some  extent  in  favor  of  a  speed  which  would  be  such 
as  to  give  the  maximum  power  under  flood  conditions.  It  is 
here  assumed  that,  whatever  speed  be  selected,  the  turbines 


TURBINE  CHARACTERISTICS  167 

must  run  at  that  constant  speed  regardless  of  fluctuations  in  the 
water.  If  the  speed  is  constant,  it  is  seen  that  (f>\^h  =  constant. 
Thus  if  the  head  decreases,  the  value  of  <£  must  increase.  In 
Fig.  109  may  be  seen  the  performance  of  a  turbine  at  a  confetant 
speed  under  different  heads.  By  making  a  different  choice  of 
the  speed  for  the  normal  "head,  the  results  under  all  other  con- 
ditions will  be  altered,  and  careful  study  must  be  made  of  all 
the  variables  to  decide  what  is  best. 

115.  Specific  Speed  an  Index  of  Type. — Both  the  elements  of 
speed  and  capacity  are  involved  in  the  specific  speed.  It  was 
stated  in  Art.  38  that  both  speed  and  capacity  were  merely 
relative  terms;  that  is,  a  high-speed  turbine  is  not  necessarily 
one  which  runs  at  a  high  r.p.m.,  but  one  whose  speed  is  high 
compared  with  other  turbines  of  the  same  power  under  the  same 
head.  In  like  manner  a  high-capacity  turbine  is  not  necessarily 
one  of  great  power  but  merely  one  whose  power  is  high  compared 
with  others  at  the  same  speed  under  the  same  head.  Since 

Ns  =  -    ,y^p'  it  is  evident  that  a  low-speed,  low-capacity  turbine 

will  be  indicated  by  a  low  value  of  N8  and  a  high-speed  high-ca- 
pacity turbine  by  a  high  value  of  Na.  As  stated  in  Art.  108, 
values  of  N9  for  the  tangential  water  wheel  may  run  up  as  high 
as  5  or  6,  for  the  reaction  turbine  they  range  from  10  to  100. 
Values  in  the  neighborhood  of  20  indicate  a  runner  such  as  Type 
I  in  Fig.  34,  while  values  in  the  neighborhood  of  80  indicate 
Type  IV.  Thus  when  the  speed  and  horsepower  of  any  turbine 
under  a  given  head  are  specified  the  type  of  turbine  necessary 
is  fixed. 

Other  things  being  equal,  it  is  seen  that  a  high  head  means  a 
comparatively  low  value  of  Na  while  a  low  head  means  a  high 
value.  Aside  from  any  structural  features  it  is  apparent  that  a 
high  head  calls  for  a  tangential  water  wheel  or  a  low-speed  re- 
action turbine,  while  a  low  head  demands  a  high-speed  reaction 
turbine.  However,  the  head  alone  does  not  determine  the  value 
of  N».  So  far  as  the  r.p.m.  is  concerned  there  may  be  consider- 
able variation,  yet  neither  a  very  low  nor  a  very  high  r.p.m. 
is  desirable  and  for  the  present  purpose  we  may  suppose  that  it 
is  restricted  within  narrow  limits.  The  value  of  N»  will  thus 
be  affected  by  the  power  of  the  turbine  as  well  as  the  head.  If 
the  head  is  high  the  value  of  Nt  may  still  be  high  enough  to 
require  a  reaction  turbine.  Or  if  the  head  is  very  low  and  the 


168  HYDRAULIC  TURBINES 

power  is  likewise  low  a  low  value  of  specific  speed  may  result.  It 
is  thus  clear  that  the  choice  of  the  type  of  turbine  is  a  function 
of  the  power  and  speed  as  well  as  the  head. 

Since  a  given  turbine  under  a  fixed  head  may  be  run  at  different 
speeds  and  gate  openings,  there  are  any  number  of  values  of  N 
and  h.p.  that  may  be  substituted  in  equation  (51),  with  a  result- 
ing variety  of  values  of  Ns  for  the  parlicular  turbine.  It  is  thus 
necessary  to  define  the  speed  and  power  for  which  this  factor  is 
to  be  computed,  if  it  is  to  have  a  definite  value  for  a  given  runner. 
The  current  practice  is  to  rate  turbines  at  the  maximum  guar- 
anteed capacity,  the  actual  maximum  capacity  being  usually 
slightly  greater  than  this,  since  the  builder  allows  a  small  margin 
to  insure  his  meeting  the  guarantee.  The  nominal  specific 
speed  is  that  corresponding  to  this  rated  capacity  at  a  stated 
speed.  But  under  a  given  head  the  turbine  speed  might  be 
selected  from  a  limited  range  of  values,  as  explained  in  Art.  114. 
It  may  be  seen  that,  though  the  true  maximum  power  of  the 
turbine  is  a  definite  value,  the  actual  maximum  power  it  can  de- 
liver at  full  gate,  under  the  operating  conditions,  depends  upon 
the  speed  at  which  it  is  run.  Hence  the  value  of  N8,  as  thus 
computed,  varies  with  the  speed,  and  is  not  a  perfectly  definite 
value.  Despite  this,  the  value  of  specific  speed  is  usually  so 
computed  because  the  rated  capacity  is  often  known  when  the 
power  and  speed  for  maximum  efficiency  are  not. 

For  accurate  comparisons  of  one  turbine  with  another  and  for 
exact  work,  it  is  best  to  select  the  values  of  power  and  speed  for 
which  the  true  maximum  efficiency  is  obtained.  The  value  of 
Ns,  so  computed,  may  be  called  the  true  specific  speed.  Since 
this  is  based  upon  a  single  definite  point,  there  can  be  but  one 
value  for  the  turbine. 

116.  Illustrations  of  Specific  Speed.— For  a  turbine  of  2000 
h.p.  at  1000  r.p.m.  under  1600  ft.  head  the  value  of  Ns  is  4.42. 
Thus  a  very  low-speed  turbine,  the  tangential  water  wheel,  is 
required.  The  actual  r.p.m.,  however,  is  high. 

For  a  5000  h.p.  turbine  at  100  r.p.m.  under  36  ft.  head  Ns 
equals  80.3.  Thus  a  high-speed  reaction  turbine  is  indicated, 
though  the  actual  r.p.m.  may  be  relatively  low. 

Suppose  that  a  12-h.p.  turbine  is  to  be  run  at  100  r.p.m.  under 
a  36-ft.  head,  the  value  of  the  specific  speed  is  3.95,  which  means 
a  tangential  water  wheel.  For  the  larger  power  under  the  same 
conditions  in  the  preceding  example  a  reaction  turbine  was  re- 


TURBINE  CHARACTERISTICS  169 

quired.     If  the  speed  were  600  r.p.m.,  however,  a  low-speed  re- 
action turbine  would  be  necessary  for  N8  would  equal  23.6. 

Suppose  that  a  20-h.p.  turbine  is  to  run  at  300  r.p.m.  under 
a  60-ft.  head.  The  value  of  N8  is  8.04  and  that  would  require 
a  tangential  water  wheel  with  two  nozzles. 

If  10,000  h.p.  is  required  at  300  r.p.m.  under  a  60-ft.  head, 
the  value  of  N8  would  be  179.5.  As  this  is  an  impossible  value 
it  would  be  necessary  to  reduce  the  speed  or  to  divide  the  power 
up  among  at  least  4  units  of  2500  h.p.  each. 

117.  Selection  of  a  Stock  Turbine. — The  choice  of  the  type 
of  turbine  will  be  taken  up  in  the  next  chapter.  For  the  present 
suppose  that  required  values  of  speed  and  power  under  the  given 
head  are  determined.  The  value  of  the  specific  speed  can  then 
be  computed  and  will  indicate  the  type  necessary.  If  the  tur- 
bine is  to  be  built  as  a  special  turbine  nothing  more  is  to  be  done 
except  to  turn  the  specifications  over  to  the  builders. 

If,  however,  the  turbine  is  to  be  selected  from  the  stock  run- 
ners listed  in  the  catalogs  of  tjie  various  makers,  it  will  be 
necessary  to  find  out  what  firms  are  prepared  to  furnish  that 
particular  type  of  runner.  It  would  be  a  tedious  matter  to 
search  through  a  number  of  tables  in  numerous  catalogs  to  find 
the  particular  combination  desired,  but  1  he  labor  is  avoided  by 
the  use  of  the  constants  given  in  the  preceding  chapter.  It  will 
be  necessary  merely  to  compute  values  of  specific  speeds  of  tur- 
bines made  by  different  manufacturers.  This  can  be  quite  readily 
done  and  such  a  table  will  always  be  available  for  future  use. 

A  make  of  turbine  should  then  be  selected  having  a  value  of 
N8  very  near  to  the  value  desired.  The  value  of  N8  ought  to 
be  as  large  as  that  required,  otherwise  the  turbine  may  prove 
deficient  in  power,  and  for  the  best  efficiency  under  the  usual 
loads  it  should  not  greatly  exceed  the  desired  value.  Having 
selected  several  suitable  runners  in  this  way,  bids  may  be  called 
for.  These  bids  should  be  accompanied  by  official  signed  re- 
ports of  Holyoke  tests  of  this  size  of  wheel  or  the  nearest  sizes  above 
and  below,  if  none  of  that  particular  size  are  available.  This 
is  to  enable  us  to  check  up  the  constants  obtained  from  catalog 
data  and  to  verify  the  efficiencies  claimed.  Holyoke  test  data 
is  very  essential  if  the  conditions  of  the  installation  are  such  that 
an  accurate  test  is  not  feasible.  In  making  a  final  choice  other 
factors  would  be  considered  such  as  efficiency  on  part  load,  and 
efficiency  and  power  under  varying  head. 


170 


HYDRAULIC  TURBINES 


118.  Illustrative  Case. — Suppose  a  turbine  is  required  to 
develop  480  h.p.  at  120  r.p.m.  under  20-ft.  head.  The  value 
of  N8  is  then  62.2.  There  are  four  makes  of  turbines  which 
approach  this  as  follows : 

TABLE  5 


Maker 

Type 

N, 

K* 

Camden  Water  Wheel  Wks.  .  . 
James  Leffel  and  Co 

United  States  Turbine  
Samson 

64.7 
61.8 

0.00190 
0.00158 

Platt  Iron  Works  

Victor  Standard  

63.0 

0.00205 

Trump  Mfg.  Co 

Standard  Trump  

61.5 

0.00210 

It  is  thus  apparent  that  any  one  of  these  manufacturers  could 
supply  a  turbine  from  their  present  designs  which  would  nearly 
fill  the  requirements.  A  number  of  other  firms  in  this  country 
could  not  fit  the  case  except  with  a  special  design  or  a  modifica- 
tion of  an  existing  design.  Thus  inspection  of  the  table  for  the 
Dayton  Globe  Iron  Works  Co.  in  Art.  110  will  show  that  the 
nearest  approach  they  have  to  it  is  their  Special  New  American 
with  an  average  value  of  Ns  of  53.7.  They  could  supply  a  tur- 
bine to  run  at  120  r.p.m.  under  the  head  specified,  but  it  would 
develop  only  358  h.p.  Or  if  they  supplied  a  turbine  capable 
of  delivering  480  h.p.  it  should  run  at  103.5  r.p.m. 

Turning  to  the  four  cases  presented  in  the  table,  it  is  apparent 
that  the  Camden  wheel  is  a  little  over  the  required  capacity, 
but  it  may  not  be  enough  to  be  objectionable.  The  Platt  ron 
Works  wheel  is  very  little  over  the  required  capacity  and  the 
Leffel  and  Trump  are  a  trifle  under  it.  If  there  is  a  little  margin 
allowable  in  the  power,  any  of  these  might  be  used.  The  value 
of  N8  according  to  which  the  wheel  is  rated  should  be  the  value 
for  the  speed  and  power  at  which  it  develops  its  best  efficiency. 
In  any  plant  the  variation  in  the  head  produces  a  deviation  from 
the  best  value  of  <j>,  if  the  wheel  be  run  at  constant  speed,  and 
thus  causes  a  drop  in  efficiency.  The  power  of  the  wheel  may 
increase  or  decrease  according  to  the  way  the  head  changes. 
Thus  in  actual  operation  the  conditions  depart  so  much  from 
those  specified  in  the  determination  of  Ns  that  small  discrep- 
ancies in  its  value  such  as  exist  in  the  table  are  of  little  im- 
portance. 

If  desired,  the  diameters  of  the  runners  may  be  determined 
by  means  of  K*  For  the  four  cases  in  the  order  given  they  will 


TURBINE  CHARACTERISTICS 


171 


1.00 


be  53.2  in.,  58.2  in.,  51.2  in.,  50.3  in.  Actually  standard  sizes 
will  not  agree  precisely  with  these  figures  and  thus  a  further 
modification  may  be  brought  about.  However,  mathematical 
exactness  must  not  be  expected  in  work  of  this  nature.  What 
we  attempt  to  do  is  merely  to  select  a  turbine  the  peak  of  whose 
efficiency  falls  as  near  as  possible  to  the  conditions  of  head, 
speed,  and  power  chosen.  Al- 
though our  conditions  may  be 
such  that  we  may  rarely  realize 
the  very  highest  efficiency  of  | 
which  the  turbine  is  capable,  *j 
yet  we  should  be  very  close  ~ 
to  it.  § 

119.  Variable  Load    and   J 
Head. — In  any  plant  the  load   ' 
is    usually    not    constant   but 
varies     over     a    considerable 
range.     In  comparing  turbines 
for  certain  situations  the  aver-    ^ 
age    operating    efficiency  may    J 
be   more  important  than  the   | 
efficiency    on    full   load   only. 
If  the  turbine  is  to  run  on  full 
load  most  of  the  time  or  if  the 
installation   is   such   that   the 
pondage  is  limited  or  lacking 
altogether,  then   the  efficiency  | 
on  part  load  is   of   little   im-  £ 
portance.     But  if   the  load  is   ~ 


=  1.00 


R.P.M.    1  Ft.  Head 
FlG.    110. 


variable  and  if  water  can  be 

stored  up  during  the  time  the 

wheel  may  be  running  under  a 

light  load  then  the  efficiency  at 

all  times  becomes  of  interest. 

If  the  plant  has  a  number  of 

units  it  is  possible  to  shut  down  some  of  them  at  times  so  as  to 

keep  the  rest  on  full  load. 

In  most  low  head  plants  the  variation  in  head  is  a  serious  item 
also  and  the  turbines  submitted  should  be  compared  as  to  their 
efficiency  and  power  under  the  range  of  head  anticipated. 

All  turbines  having  the  same  specific  speed  are  not  necessarily 


172 


HYDRAULIC  TURBINES 


loo jj  artQ  Japun'fflM' 


C>  IA  O  tO  O  "3  O  iOOOOt<3OO»u3 

dO          S         £  «5          S          to          trt  ^f          ^l         1         t         1          M.          *1          rt.          ^ 


TURBINE  CHARACTERISTICS  173 

equally  well  suited  for  the  same  conditions.  A  detailed  study  of 
the  characteristics  of  each  one  is  essential  before  the  final  choice 
can  be  made.  In  many  cases  the  best  efficiency  will  be  the 
deciding  factor.  In  others  the  average  operating  efficiency  will 
be  more  important,  and  sometimes  capacity  under  varying  head 
will  be  the  chief  item. 

These  factors  can  be  studied  by  means  of  curves  such  as  are 
shown  in  Fig.  110.  Efficiency,  discharge,  and  power  for  various 
gate  openings  reduced  to  1-ft.  head  are  plotted  against  (f>  or  the 
r.p.m.  under  1-ft.  head.  The  normal  speed  and  power  should  be 
that  corresponding  to  the  maximum  efficiency.  If  the  wheel  is 
run  at  constant  speed  a  variation  in  head  causes  a  change  in  <£. 

120.  Characteristic  Curve. — For  a  thorough  study  of  a  turbine 
the  characteristic  curve  is  a  most  valuable  graphic  aid.  The 
coordinates  of  such  a  curve  are  discharge  under  1-ft.  head 
and  0  or  r.p.m.  under  1-ft.  head.  Values  of  the  horsepower 
input  under  1-ft.  head  should  also  be  laid  off  to  correspond 
to  the  values  of  the  discharge.  Lines  should  then  be  drawn  on 
the  diagram  to  indicate  the  relation  between  speed  and  discharge 
for  various  gate  openings.  Alongside  of  each  experimental 
point  giving  this  relation,  the  value  of  the  efficiency  should  be 
written.  When  a  number  of  such  points  are  located,  lines  of 
equal  efficiency  may  be  drawn  by  interpolation. 

Another  very  good  rnethod  is  to  draw  curves  of  efficiency  as  a 
function  of  0  for  each  gate  opening.  For  any  iso-efficiency  curve 
desired  on  this  diagram  it  is  possible  to  read  off  corresponding 
values  of  <£. 

If  desired,  lines  of  equal  power  may  also  be  constructed.  To 
do  so,  assume  the  horsepower  of  the  desired  curve,  then  compute 
the  horsepower  input  for  any  efficiency  by  the  relation,  horse- 
power input  =  horsepower  output  -f-  e.  This  value  of  e  on  one 
of  the  iso-efficiency  curves  together  with  the  value  of  horse- 
power input  locates  one  or  two  points  of  an  iso-power  curve. 

The  characteristic  curve  for  a  24-in.  tangential  water  wheel  is 
shown  in  Fig.  111.  This  curve  covers  all  the  possible  conditions 
under  which  the  wheel  might  run.  The  only  way  to  extend  the 
field  would  be  to  put  on  a  larger  nozzle.  Since  the  discharge  of  a 
tangential  water  wheel  is  independent  of  the  speed  the  lines  for 
the  various  gate  openings  will  be  straight.  For  the  reaction 
turbine  they  will  be  curved  as  seen  in  Fig.  112.  The  latter  is  a 
portion  of  a  chracteristic  curve  for  a  high-speed  turbine. 


174 


HYDRAULIC  TURBINES 


Any  marked  irregularities  in  the  characteristic  curve  are 
indications  of  errors  in  the  test.  It  is  possible  for  there  to  be 
only  one  peak  in  the  efficiency  curves  and  an  indication  of  two, 
as  sometimes  occurs,  is  due  to  incorrect  data. 

This  method  of  plotting  test  data  and  also  that  shown  in  Fig. 
110  were  first  given  by  Prof.  D.  W.  Mead  in  his  "  Water  Power 
Engineering."  Other  diagrams  have  also  been  proposed  by 
various  men,  the  object  of  all  being  to  represent  the  fundamental 
variables  in  the  best  form  for  the  ready  comparison  of  one  turbine 
with  another. 


1.4 


H.E»  Input  under  One  Foot  Head 
1.6  1.8  2.0  .          2.2  2.4 


2.6 


1.00  — 


0.95  = 


0.70  - 


10      11      12      13      14       15       16      17       18      19       20      21      22      23      24      25 

Discharge?  in  Cu.Ft.  per  Sec.  under  One  Foot  Head,. 
FIG.  112. — Characteristic  curve  for  a  high  speed  reaction  turbine. 

121.  Use  of  Characteristic  Curve. — From  the  characteristic 
curve  it  is  apparent,  at  a  glance,  at  what  speed  the  turbine  should 
.run  for  the  best  efficiency  at  any  gate  opening.  The  best  effi- 
ciency in  Fig.  Ill  is  obtained  when  $  =  0.457  or  Ni  =  34,  and 
with  the  needle  open  6  turns.  With  full  nozzle  opening  the  best 
value  of  Ni  is  35,  with  the  needle  open  3  turns  the  best  speed  is 
such  that  Ni  =  32.  (With  the  reaction  turbine  these  differences 
would  be  greater.) 

From  the  characteristic  curves  any  other  curves  may  be  con- 
structed. For  constant  speed  follow  along  a  horizontal  line,  for 
a  fixed  gate  opening  follow  ^long  the  curve  for  that  relation. 

If  it  is  desired  to  investigate  the  effect  of  change  of  head  when 


TURBINE  CHARACTERISTICS  175 

the  speed  is  kept  constant,  compute  the  new  values  of  <£  or  Ni. 
Thus  the  curve  in  Fig.  Ill  was  determined  by  a  test  under  a  head 
of  65.5Jft.  and  the  best  speed  was  275  r.p.m.  That  corresponded 
to  <f>  =  0.457  or  Ni  =  34.  (The  value  of  D  used  for  computing  <£ 
was  slightly  different  from  the  nominal  diameter.)  If  the  speed 
is  maintained  at  275  r.p.m.  when  the  head  is  74  ft.,  then  </>  =  0.429 
and  Ni  =  32.  If  h  =  55  ft.,  0  =  0.497  and  Ni  =  37.  In  the 
last  case  the  best  efficiency  would  be  77  per  cent.,  a  drop  of  1  per 
cent. 

The  iso-efficiency  curves  represent  contour  lines  on  a  relief 
model  and  thus  the  point  of  maximum  efficiency  is  represented 
by  a  peak.  It  is  apparent  that  for  varying  loads  or  heads  a  tur- 
bine giving  a  diagram,  that  indicates  a  model  with  gentle  slopes 
from  this  point,  would  probably  be  better  than  a  turbine  for 
which  the  peak  might  be  higher  and  the  slopes  steeper.  The  rela- 
tiVe  increase  in  discharge  capacity  at  full  gate  as  0  increases  is  also 
apparent  and  indicates  which  turbine  is  better  for  operation  at 
reduced  head  but  normal  speed. 

122.  QUESTIONS  AND  PROBLEMS 

1.  For  any  turbine,  how  does  the  speed  for  highest  efficiency  vary  with 
the  gate  opening  used?     How  does  the  efficiency  vary  with  the  gate  opening 
for  any  speed?     At  what  speed  and  gate  will  maximum  efficiency  be  found, 
as  compared  with  maximum  power? 

2.  Should  a  turbine  necessarily  be  run  at  the  speed  for  maximum  effi- 
ciency?    Why? 

3.  What  happens  to  the  power  and  efficiency  of  a  turbine  when  the  head 
changes,  but  the  speed  is  kept  constant?     In  time  of  flood,  what  is  the  im- 
portant consideration? 

4.  What  is  the  difference  between  the  true  and  the  nominal  specific 
speed?     What  would  be  the  general  profile  of  a  runner  whose  specific  speed 
was  10?     What  of  one  whose  specific  speed  was  100? 

6.  When  is  efficiency  on  full  load  important  and  when  is  efficiency  on 
part  load  of  more  value?  When  is  maximum  power  of  principal  interest? 
When  is  maximum  speed  the  chief  object? 

6.  If  a  plant  contains  a  number  of  units,  what  should  be  done  if  all  of 
them  are  carrying  half  load?     Why?     Would  there  be  any  object  in  shut- 
ting down  some  of  them  if  the  supply  of  water  was  abundant? 

7.  If  there  is  a  great  shortage  of  water  so  that  the  supply  is  inadequate 
for  all  the  wheels  at  full  head,  so  that  the  water  level  falls  considerably  below 
normal  before  equilibrium  is  attained,  is  it  better  to  operate  the  plant  with 
all  the  wheels  or  shut  down  enough  of  them  to  keep  the  water  level  near  the 
crest  of  the  spillway?     Would  it  be  better  to  shut  down  some  of  the  wheels 
to  accomplish  this  or  to  operate  all  of  them  at  part  gate  opening? 


176  HYDRAULIC  TURBINES 

8.  In  Fig.  112,  what  is  the  maximum  efficiency  and  what  is  the  value  of 
the  efficiency  for  maximum  power?     What  is  the  ratio  of  the  speeds  for 
each  of  these,  and  what  is  the  ratio  of  the  power  outputs  at  full  gate  for 
each  of  these  speeds? 

9.  If  the  turbine,  whose  performance  is  shown  by  Fig.  112,  is  run  at  the 
speed  for  best  efficiency  under  a  head  of  50  ft.,  what  will  be  its  maximum 
output  and  what  the  power  at  the  point  of  maximum  efficiency?     If  the 
speed  is  kept  at  this  same  value  while  the  head  falls  to  36.5  ft.,  what  will 
be  the  value  of  the  maximum  power  delivered? 

Ans.     755  h.p.,  720  h.p.,  498  h.p. 

10.  In  problem  (9)  the  second  value  of  the  head  is  73  per  cent,  of  its 
initial  value  and  the  maximum  power  is  66  per  cent,  of  its  value  in  the  first 
case.     For  the  impulse  turbine  in  Fig.  Ill,  what  would  be  the  ratio  of  the 
maximum  power  outputs,  if  the  head  dropped  the  same  proportional  amount 
while  the  speed  remained  the  same  as  for  the  maximum  efficiency  under  the 
initial  head?  Ans.     58.7  per  cent. 

11.  Suppose  that  an  impulse  wheel,  similar  to  that  for  which  the  curves 
of  Fig.    Ill,   were  drawn,  is  made  of  such  a  size  as  to  develop  5000  h.p. 
under  a  head  of  1200  ft.     Find  the  diameter  of  the  wheel,  its  r.p.m.,  and 
plot  a  curve  between  efficiency  and  power  for  a  constant  speed. 


CHAPTER  'XIII 
SELECTION  OF  TYPE  OF  TURBINE 

123.  Possible  Choice. — It  has  been  shown  that,  if  the  speed 
and  power  under  a  given  head  are  fixed,  the  type  of  turbine 
necessary  is  determined.     If  there  is  some  leeway  in  these  mat- 
ters it  may  be  possible  to  vary  the  specific  speed  through  a 
considerable  range  of  values.     Suppose  turbines  of  a  given  power 
may  be  run  at  120  r.p.m.,  at  600  r.p.m.,  or  at  900  r.p.m.     Each 
one  of  these  would  give  us  a  different  specific  speed  and  thus  a 
different  type  of  runner.     Or,  if  the  speed  be  fixed,  the  power, 
such  as  20,000  h.p.  may  be  developed  in  a  single  unit,  in  two 
units  of   10,000  h.p.  each,  or  in  eight  units  of  2500  h.p.  each. 
Again  we  have  different  types  of  runners  demanded.     Both  the 
speed  and  power  may  be  varied  in  some  cases  and  the  choice  is 
wider   still.     As    an  example,   it  may  be  required  to  develop 
500  h.p.  under  140-ft.  head.     Suppose  this  power  is  to  be  divided 
up  between  two  runners  and  the  speed  to  be  120  r.p.m.     The 
value  of  Ns  is  then  4.12,  showing  that  a  double  overhung  tan- 
gential water  wheel  is  required.     Or  if  the  power  be  developed 
in  a  single  runner  at  600  r.p.m.,  the  value  of  N8  would  be  29.2, 
which  would  call  for  a  reaction  turbine. 

It  is  customary  to  choose  a  speed  between  certain  limits,  as 
neither  a  very  low  nor  a  very  high  r.p.m.  is  desirable.  Also  the 
number  of  units  into  which  a  given  power  is  divided  is  limited. 
Nevertheless  considerable  latitude  is  left.  It  remains  to  be  seen 
what  considerations  would  lead  us  to  choose  such  values  of  speed 
and  power  as  would  permit  the  use  of  a  certain  type  of  runner. 

124.  Maximum    Efficiency. — The    best    efficiency    developed 
by  a  turbine  will  depend,  to  some  extent,  upon  the  class  to  which 
it  belongs.     The  impulse  and  reaction  turbines  are  so  different 
in  their  construction  and  operation  that  the  difference  in  effi- 
ciency between  them  can  be  determined  solely  by  experiment. 
However,  abstract  reasoning  alone  will  lead  to  certain  conclusions 
as  to  the  relative  merits  of  different  types  within  each  of  these 
two  main  divisions. 

12  177 


178  HYDRAULIC  TURBINES 

For  the  tangential  water  wheel  it  has  been  shown  that,  if  the 
highest  efficiency  is  to  be  obtained,  certain  proportions  must  not 
be  exceeded.  If  we  desire  a  specific  speed  higher  than  4,  it  is 
necessary  to  pass  beyond  these  limits  and  thus  a  wheel  whose 
specific  speed  is  as  high  as  5  or  6  will  not  have  as  high  an  efficiency 
as  the  normal  type.  On  the  other  hand  too  low  a  specific  speed 
is  not  conducive  to  efficiency,  since  the  diameter  of  the  wheel 
becomes  relatively  large  in  proportion  to  the  power  developed, 
so  that  the  bearing  friction  and  windage  losses  tend  to  become 
too  large  in  percentage  value.  The  value  of  Ns  for  the  highest 
efficiency  is  about  4. 

A  low  specific  speed  reaction  turbine,  such  as  Type  I  in  Fig.  34 
for  example,  will  have  a  small  value  of  the  angle  a\.  A  considera- 
tion of  the  theory,  especially  equation  (33),  shows  that  this  is 
conducive  to  high  efficiency.  However  this  is  more  than  offset 
by  other  factors,  such  as  the  large  percentage  value  of  the  disk 
friction,  as  explained  in  Art.  108.  In  addition,  the  leakage  area 
through  the  clearance  spaces  becomes  a  greater  proportion  of 
the  area  through  the  turbine  passages,  and  also  the  hydraulic 
friction  through  the  small  bucket  passages  is  larger.  The  result 
of  all  these  factors  is  that  the  efficiency  tends  to  be  reduced  as 
very  small  values  of  the  specific  speed  are  approached. 

A  medium  specific  speed  turbine  runner  would  have  a  some- 
what larger  value  of  the  guide  vane  angle  but  this  slight  dis- 
advantage would  be  more  than  offset  by  the  reduction  in  the 
relative  values  of  the  disk  friction,  leakage  loss,  and  hydraulic 
friction  loss  within  the  runner.  Thus  this  type  would  have  a 
higher  efficiency  than  the  former. 

But  when  the  high  specific  speed  type  is  reached  the  inherently 
large  value  of  the  discharge  loss  is  such  as  to  materially  reduce 
the  efficiency.  This  reduction  is  aided  also  by  the  large  value 
given  to  the  guide  vane  angle  and  opposed  by  the  decreased  disk 
friction,  leakage  through  the  clearance  spaces,  and  internal 
hydraulic  friction.  However  the  effect  of  these  latter  factors 
is  not  sufficient  to  offset  the  increased  discharge  loss.  In  other 
words,  efficiency  has  been  sacrificed  in  favor  of  increased  speed 
and  capacity,  just  as  in  the  case  of  a  high-speed  impulse  turbine. 

This  reasoning  is  borne  out  by  the  facts,  as  can  be  seen  by  Fig. 
113,  where  efficiency  is  plotted  as  a  function  of  specific  speed.1 
A  number  of  test  points  were  located  and  the  curve  shown  was 

i  L.  F.  Moody,  Trans.  A.  S.  C.  E.,  Vol.  LXVI,  p.  347. 


SELECTION  OF  TYPE  OF  TURBINE 


179 


drawn  through  the  highest  on  the  sheet.  It  shows  what  has 
actually  been  accomplished  and  it  also  shows  how  the  maximum 
efficiency  varies  with  the  type  of  turbine.  It  is  apparent  that  if 
one  desires  the  highest  efficiency  possible  a  specific  speed  should 
be  chosen  between  25  and  50. 

It  must  not  be  thought  that  this  curve  represents  the  results 
that  one  should  expect  in  every  case.  It  merely  shows  the  rela- 
tive merits  of  the  different  types.  The  actual  efficiency  obtained 
depends  not  only  upon  the  specific  speed  but  also  upon  the  ca- 
pacity of  the  turbine  and  the  head  and  other  factors.  The  larger 
the  capacity  of  a  turbine  the  higher  the  efficiency  will  be.  In  a 
given  case  the  efficiency  obtained  for  a  specific  speed  of  30,  say, 
might  be  only  83  instead  of  the  93  shown  by  the  curve.  But  if 
the  specific  speed  had  been  95  instead  of  30  the  efficiency  realized 
might  have  been  only  73. 


20 


30  40  50  60  70 

Values  of  the  Specific  Speed  -  M8 

FIG.  113. 


80 


100 


Higher  efficiencies  have  been  attained  with  reaction  turbines 
than  with  Pelt  on  wheels.  The  maximum  recorded  efficiency 
for  the  former  is  93.7  per  cent,  and  quite  a  few  large  units 
have  shown  efficiencies  over  90  per  cent,  where  conditions  were 
favorable.  The  highest  reported  value  for  an  impulse  wheel 
is  V89  per  cent,  but  the  usual  maximum  is  about  82  per  cent. 
However  the  efficiency  of  a  reaction  turbine  is  a  function  of 
its  capacity,  that  is  for  small  sizes  the  efficiency  is  relatively 
low.  As  the  larger  sizes  are  reached  this  difference  dis- 
appears. The  reason  for  this  is  that  the  clearance  spaces 
and  hence  leakage  losses  are  a  greater  percentage  with  the 
small  sizes.  The  efficiency  of  the  Pelton  wheel  is  not  depend- 
ent on  its  size.  Hence  for  smaller  powers  the  tangential  wheel 
may  have  a  higherj[maximum  efficiency  than  the  reaction 
turbine. 

125.  Efficiency  on  Part-load. — Full-load  will  be  defined  as  the 
load  under  which  a  turbine  develops  its  maximum  efficiency. 


180  HYDRAULIC  TURBINES 

Anything  above  that  will  be  called  an  overload  and  anything 
less  than  that  will  be  known  as  part-load. 

It  has  already  been  explained  (Art.  84)  that  to  obtain  the  high- 
est efficiency  the  water  must  enter  without  turbulent  vortex 
motion  (known  as  shock)  and  must  leave  with  as  little  velocity 
as  possible.  In  order  to  obtain  the  former  the  vane  angle  jS'i 
must  agree  with  the  angle  of  the  relative  velocity  of  the  water  as 
determined  by  the  vector  diagram,  and  the  quantity  of  water 
should  be  such  that  its  relative  velocity  t/i,  as  determined  by  the 
equation  of  continuity,  should  agree  with  the  velocity  v\y  as 
determined  by  the  vector  diagram  of  velocities.  In  order  to  re- 
duce the  discharge  loss  to  a  minimum  it  has  also  been  shown  that 
a2  should  have  a  value  of  approximately  90°. 

There  is  practically  no  additional  loss  at  entrance  to  the  buck- 
ets of  a  Pelton  wheel  due  to  the  reduction  in  the  size  of  the  jet 
at  part-load.  If  the  jet  and  wheel  velocities  remained  just  the 
same,  the  velocity  diagrams  would  be  identical  at  all  loads. 
Actually  the  jet  velocity  may  vary  slightly  but  the  shape  of  the 
buckets  is  such  that  there  is  no  well  denned  vane  angle  at  en- 
trance. And  since,  in  the  impulse  turbine,  the  relative  velocity 
through  the  runner  is  not  determined  by  the  equation  of  con- 
tinuity, there  can  be  no  abrupt  change  in  either  the  direction  or 
magnitude  of  the  relative  velocity  of  the  water  at  entrance.  But 
this  is  not  the  case  with  the  reaction  turbine.  The  smaller  gate 
opening  changes  the  angle  a'\.  This  alters  the  entrance  velocity 
diagram.  Hence  the  angle  /3\  will  no  longer  agree  with  the  vane 
angle  0'i.  Since  the  quantity  of  water  discharged  per  unit  time 
is  less  than  before,  it  follows  that  the  velocity  t/i,  as  determined 
by  the  area  of  the  runner  passages,  is  less  than  the  value  at  full- 
load.  Thus  when  a  reaction  turbine  runs  at  part-gate  there  are 
eddy  losses  produced  at  entrance  to  the  runner  due  to  the  abrupt 
change  in  the  direction  and  magnitude  of  the  velocity  of  the 
water  through  the  wheel  passages.  No  such  losses  occur  with  the 
impulse  turbine. 

At  the  point  of  discharge  the  velocity  diagram  for  the  tangential 
wheel  is  practically  the  same  at  all  loads,  provided  the  jet  velocity 
and  bucket  velocity  are  the  same.  f  There  may  be  slight  increases 
in  the  losses  in  flow  over  the  bucket  surfaces  which  would  affect 
this  statement  somewhat  for  very  large  or  very  small  nozzle 
openings,  but  for  a  reasonable  range  the  statement  is  true.  Thus 
the  discharge  loss  would  be  the  same  at  all  loads.  But  for  the 


SELECTION  OF  TYPE  OF  TURBINE  181 

reaction  turbine,  since  the  water  completely  fills  the  bucket  pas- 
sages, a  reduced  rate  of  discharge  requires  a  proportionate  reduc- 
tion in  the  relative  velocity  vz.  Thus  F2  and  hence  the  discharge 
loss  are  inevitably  increased.  The  higher  the  specific  speed  of 
the  turbine  the  greater  the  discharge  loss  .at  the  normal  gate 
opening,  and  hence  the  greater  the  effect  produced  upon  the 
efficiency  when  this  loss  is  increased  at  part-gate. 

It  is  thus  apparent  that  at  part-load  there  are  inherent  losses 
.within  the  reaction  turbine  that  are  not  found  with  the  Pelton 
wheel. 

In  fact  the  hydraulic  efficiency  of  the  latter  would  appear 
to  be  the  same  at  all  nozzle  openings.  In  reality  the  re- 
duction in  the  velocity  coefficient  of  the  nozzle,  as  the  needle 
closes  the  discharge  area,  together  with  some  change  in  the  bucket 
friction,  changes  the  efficiency  slightly."  It  is  the  gross  efficiency 
with  which  we  are  really  concerned,  and  of  course  the  mechan- 
ical losses  due  to  friction  and  windage,  which  are  constant  at 
constant  speed,  cause  the  efficiency  to  decrease  as  the  gate 
opening  decreases.  But  the  efficiency-load  curve  of  the  tangen- 
tial water  wheel  is  inherently  a  flat  curve. 

The  losses  within  the  reaction  turbine  runner  are  such  that  the 
hydraulic  efficiency  must  decrease  as  the  gate  is  changed  in 
either  direction  from  the  position  at  full-load.  Hence  the  effi- 
ciency at  part-load  or  overload  tends  to  be  less  than  that  for 
the  impulse  wheel,  as  shown  in  Fig.  114  (assuming  both  to  be  the 
same  at  full-load),  and  the  higher  the  specific  speed  the  steeper 
will  the  efficiency  curve  be. 

for  the  tangential  water  wheel  in  Fig.  Ill  it  can  be  seen  that 
the  best  speed  is  slightly  different  for  different  gate  openings 
and  that  it  increases  as  the  latter  increases.  This  is  also  true 
with  the  reaction  turbine,  but  in  a  mofeTmarked  degree  as  can 
be  seen  in  Fig.  112.  If  the  speed  is  selected  so  as  to  give  the  best 
efficiency  at  a  certain  gate  opening  it  will  not  be  correct  for  any 
other  gate  opening  and  thus  efficiency  will  be  sacrificed  at  ail 
gates  except  one. 

This  variation  of  the  best  speed  with  different  gate  openings 
is  found  in  all  turbines,  but  not  in  the  same  degree.  With  the  low- 
speed  reaction  turbine  it  is  small,  approaching  the  tangential 
water  wheel  in  that  regard.  With  the  high-speed  reaction  tur- 
bine it  is  very  marked.  There  seems  to  be  little  difference  be- 


182 


HYDRAULIC  TURBINES 


tween  turbines,  in  this  regard,  for  specific  speeds  less  than  50; 
but  for  specific  speeds  above  that,  it  increases  rapidly.1 

If,  then,  a  constant  speed  be  selected  which  is  the  best  for  full- 
load,  there  will  be  a  sacrifice  of  efficiency  on  part-load,  and  this 


Load 
FIG.  114. — Relative  efficiencies  on  part-load  of  impulse  and  reaction  turbines. 

100 1 


10       20 


50      60       70       80 
Percent  of  Full  Load 

FIG.  115. — Typical  efficiency  curves. 


100     110     120     130 


sacrifice  will  be  greater  the  higher  the  specific  speed  of  the  tur- 
bine.    These  considerations,  together  with  the  facts  given  in  the 
preceding  article,  imply  efficiency  curves  for  the  various  types 
i  C.  W.  Larner,  Trans.  A.  S.  C.  E.,  Vol.  LXVI,  p.  341  (1910). 


SELECTION  OF  TYPE  OF  TURBINE 


183 


such  as  are  shown  in  Fig.  115.  To  prevent  confusion  the  effi- 
ciency curve  for  a  low-speed  turbine  is  not  shown,  but  its  effi- 
ciency on  full-load  would  be  about  the  same  as  that  for  the 
tangential  water  wheel,  while  on  part-load  it  would  be  a  little 
less. 

It  may  also  be  noticed  that  there  is  less  overload  capacity 
with  the  high-speed  turbine  than  with  the  other  types.  This 
is  because  the  point  6f  maximum  efficiency  is  nearer  full-gate 
than  with  the  other  types.  If  the  customary  25  per  cent,  over- 
load must  be  allowed,  then  the  normal  load  must  be  less  than 
the  power  for  maximum  efficiency  with  a  further  decrease  in 
operating  efficiency. 


0.2 


0.4 


1.0 


1.2 


1.4 


0.6  0.8 

Gate  Opening 

FIG.  116. — Relation  between  power  and  gate  opening  for  same  speed  under  differ- 
ent heads. 

Thus  from  the  tangential  wheel  on  the  one  hand  to  the  high- 
speed reaction  turbine  on  the  other  the  relative  efficiency  on  part- 
load  decreases  as  the  specific  speed  increases. 

126.  Overgate  with  High-speed  Turbines.— With  the  wicket 
or  swing  gates,  as  used  today,  there  is  no  definite  limit  to  their 
opening  save  that  imposed  by  an  arbitrary  mechanical  stop.  As 
the  gate  opening  increases  the  rate  of  discharge  and  hence  the 
power  of  the  turbine  increases,  as  shown  by  curve  for  <£  =  0.78  in 
Fig.  116.  But  with  too  great  an  angle  of  the  vanes  the  efficiency 
decreases  so  much  that  the  power  output  no  longer  continues  to 
increase  and  may  even  decrease.  Ordinarily  there  is  no  advan- 
tage gained  by  opening  the  gates  any  wider  than  that  necessary 
to  secure  maximum  power,  and  hence  the  mechanism  is  usually 
so  constructed  that  it  cannot  move  the  gates  any  farther  than 


184 


HYDRAULIC  TURBINES 


this  position.  This  may  be  termed  "  full-gate  op'ening."1  If  the 
construction  is  such  that  the  gates  can  be  opened  wider  than  this 
amount,  the  range  from  full-gate  up  to  the  maximum  opening  may 
be  termed  " over-gate." 


70 


1.3 


r 


0. 


0,5 


Values  of  0 
FIG.  117. — Characteristics  of  low-speed  runner.     N8  =  27. 

If  the  normal  speed  be  taken  as  the  speed  at  which  the  wheel 
develops  its  maximum  efficiency,  it  may  be  seen  in  Fig.  112  that 

1  This  is  a  purely  arbitrary  definition,  but  there  is  at  present  no  agreement 
as  to  what  the  term  "full-gate"  really  signifies,  and  so  it  will  be  here  used 
as  defined  above.  It  may  be  noted  that  the  gate  movement  might  also  be 
limited  to  something  less  than  the  position  shown,  and  in  such  an  event  it 
would  be  logical  to  denote  the  maximum  opening  as  the  "  full-gate."  The 
effect  of  this  construction  would  be  to  decrease  the  overload  capacity  or  to 
move  the  power  for  maximum  efficiency  nearer  to  the  maximum  power. 
For  the  same  maximum  power  this  would  require  a  slightly  larger  runner. 


SELECTION  OF  TYPE  OF  TURBINE     ,       185 

the  power  at  full-gate  increases  as  the  speed  increases  above  nor- 
mal. This  is  a  peculiarity  of  the  high-speed  turbine.  With  the 
medium  and  low-speed  turbine  there  is  no  such  increase.  In 
fact,  as  shown  in  Fig.  117,  there  may  be  a  reduction  in  power 
with  the  low  specific  speed  turbine  at  full-gate,  if  the  speed  is 
increased  above  normal.  The  explanation  of  this  difference 
in  the  two  types  is  that  with  the  inward  flow  turbine  the  centrifugal 
action  opposes  the  flow  of  water,  and  hence  the  rate  of  discharge 
tends  to  decrease  as  the  speed  increases,  while  with  the  outward 
flow  turbine  the  centrifugal  action  tends  to  increase  the  rate  of 
discharge  with  the  speed.  This  may  be  seen  in  Fig.  95,  page  126. 
The  low  specific  speed  runner  approaches  the  pure  radial  inward 
flow  type,  while  the  high  specific  speed  runner  of  the  present  with 
inward,  downward,  and  outward  flow  (the  radius  to  the  outer  limit 
of  the  discharge  edge  being  often  as  much  as  one-third  greater 
than  the  radius  to  the  entrance  edge)  approaches  the  outward 
flow  turbine  in  this  characteristic.  Thus,  despite  the  decrease 
in  efficiency,  as  the  speed  departs  from  the  normal,  the  in- 
creased rate  of  discharge  tends  to  increase  the  power  output  for 
a  certain  range  of  speed  above  normal.  This  feature  of  the 
high-speed  turbine  is  of  great  value,  as  it  especially  fits  it  for 
the  class  of  service,  to  which  it  is  otherwise  adapted. 

As  has  been  explained,  the  maximum  opening  of  the  turbine 
gate  would  usually  be  that  at  which  no  further  increase  in  power 
at  normal  speed  could  be  obtained  and  this  is  termed  " full-gate." 
But  with  the  high  specific  speed  turbine  it  is  found  that,  when 
running  at  a  speed  above  normal,  the  power  continues  to  increase 
for  an  opening  of  the  gate  beyond  its  usual  maximum  value,  as 
shown  by  curves  for  </>  =  1.03  and  1.10  in  Fig.  116.  A  turbine 
so  constructed  that  the  gate  can  be  opened  wider  than  the  maxi- 
mum value  necessary  under  normal  conditions  is  said  to  be 
"overgated. "  This  additional  gate  opening  would  be  of  no 
value  with  a  low-speed  turbine  under  any  circumstances,  and  it 
would  be  of  no  value  with  a  high-speed  turbine  under  normal 
conditions.  But,  not  only  does  the  power  of  the  latter  increase 
at  full-gate  for  speeds  higher  than  the  normal,  but  by  opening 
the  gate  wider  than  the  usual  value  the  power  may  be  still 
further  increased  as  may  be  seen  in  Figs.  107  and  118.  The 
nominal  full-gate  opening  is  denoted  by  unity. 

If  the  head  on  a  water  power  plant  decreases,  as  in  time  of 
flood,  the  capacity  of  each  turbine  is  reduced,  with  a  resulting 


186 


HYDRAULIC  TURBINES 


shortage  of  power.  If  the  wheels  must  run  at  a  constant  speed, 
as  is  usually  the  case,  the  speed  is  no  longer  correct  for  the  head, 
and  this  causes  a  decrease  in  efficiency  with  a  further  reduction 
of  power.  »j 

Any  feature  which  will  improve  the  capacity  of  the  turbine 
under  these  circumstances  is  of  value.  Now  since  speed  is 
proportional  to  0VX  a  constant  speed  under  a  reduced  head 
means  an  increase  in  <f>  above  its  normal  value.  As  has  been  seen, 


47       49 


51       53        55       57       59 
R.P.M.-l  1't.  Head 
FIG.   118. — High-speed  turbine 


61       63       65      67 


an  increase  in  <£  above  its  normal  value  causes  no  increase  in  the 
power  of  a  low-  or  medium-speed  runner,  but  with  the  high-speed 
runner  not  only  does  the  power  at  full-gate  increase  but,  by 
overgating,  the  power  may  be  still  further  increased.  The  effect 
of  an  overgate  is  to  materially  increase  the  capacity  of  the  tur- 
bine at  a  time  when  there  is  a  shortage  of  power.  This  feature  is 
not  possessed  by  lower  speed  runners. 

The  high  specific  speed  turbine  has  a  lower  maximum  efficiency, 


SELECTION  OF  TYPE  OF  TURBINE 


187 


a  lower  part  load  efficiency,  and  less  overload  capacity  than  the 
medium  speed  turbine  when  both  are  operated  under  a  constant 
head,  but  its  higher  speed  is  a  decided  advantage  where  the  nor- 
mal head  is  low.  But  it  is  with  low  head  plants  that  flood  con- 
ditions are  most  serious  in  their  effects  upon  the  capacity  of  the 
plant  and,  as  has  been  seen,  the  characteristics  of  the  high-speed 
runner  are  such  that  it  is  able  to  deliver  more  power  under  these 
circumstances. 

The  results  obtained  with  "overgating"  high-speed  turbines 
may  be  seen  in  Figs.  107  and  118.  In  the  latter  case  full-gate 
is  denoted  by  1.00  and  the  maximum  gate  opening  by  1.077. 
The  normal  speed  is  52  r.p.m.  under  1  ft.  head.  At  that  speed 
any  further  gate  opening  would  be  of  no  advantage,  and  in  fact 
would  merely  cause  a  drop  in  efficiency.  But  at  a  higher  speed, 
such  as  65  r.p.m.,  the  overgate  feature  raises  the  power  under  1ft. 
head  from  2.0  h.p.  at  full-gate  to  2.35.  Bearing  in  mind  that  a 
medium  speed  runner  for  the  same  situation  would  deliver  less 
than  2  h.p.  under  these  circumstances,  it  is  seen  how  much  supe- 
rior the  high  specific  speed  turbine  is  for  the  particular  condi- 
tions of  service. 

The  differences  between  the  low-  and  high-speed  runners  are 
brought  out  in  the  following  table.  The  normal  head  is  15  ft. 
and  the  wheels  develop  100  h.p.  In  time  of  high  water  the  head 
will  decrease  to  10  ft.,  while  the  wheels  are  kept  at  their  normal 
speed. 

TABLE  6. — HEAD  =  15  Ft. 


Type 

Ar, 

R.p.m. 

4 

Gate 

H.p. 

Efficiency 

Low  speed 

35.2 

104 

0.70 

1.00 

100 

85 

High  speed 

77  7 

232 

0  81 

1  00 

100 

82 

TABLE  7.— HEAD  =  10  Ft. 


Type 

JV.» 

R.p.m. 

<t> 

Gate 

H.p. 

Efficiency 

Low  speed 

35  2 

104 

0  86 

1.00 

47  3 

78 

High  speed  

77.7 

232 

0.99 

1.00 

49.3 

76 

High  speed  

232 

0.99 

1.077 

58.0 

80 

1  These  values  of  specific  speed  apply  only  when  the  turbine  is  developing 
its  best  efficiency.  Under  the  reduced  head  with  an  incorrect  value  of  <f>, 
the  real  specific  speed  is  different.  But  as  an  index  of  the  type  the  values 
given  are  always  appropriate. 


188 


HYDRAULIC  TURBINES 


By  means  of  the  overgate  feature  the  high-speed  turbine  is  seen 
to  be  capable  of  developing  22J^  per  cent,  more  power  under  the 
lower  head  than  the  low-speed  turbine.  The  efficiency  is  also 
seen  to  be  greater. 

127.  Type  of  Runner  as  a  Function  of  Head. — It  has  been 
stated  that  the  choice  of  the  type  of  turbine  is  a  function  of  the 
power  and  the  speed  desired,  as  well  as  the  head.  While  this  is 
true,  the  value  of  the  head  does  exert  a  predominating  influence 
and  hence  there  is  some  justification  for  the  presentation  of  a 
relationship  between  the  two,  such  as  is  given  in  Table  8.  How- 
ever, it  should  be  noted  that  the  figures  given  for  specific  speed 
are  merely  limits.  Thus  a  head  of  100  ft.  does  not  require  a 
turbine  whose  specific  speed  is  50  for  example.  The  latter 
is  merely  the  maximum  value  found  in  current  practice  for  such  a 
head,  and  a  lower  value  of  Ns  might  be  used.  Within  this  max- 
imum limit  the  specific  speed  chosen  would  depend  upon  the 
power  and  speed  and  a  consideration  of  the  characteristics  desired. 

TABLE  8. — RELATION  BETWEEN  HEAD  AND  SPECIFIC  SPEED 


Head   ft. 

Maximum 
value  of  Na 

Type  of  setting 

20 

100 

25 

90 

Vertical  shaft  single  runner  units  with 

35 

80 

good  draft  tubes. 

50 

70 

65 

60 

Single  runner,  either  horizontal 

or  ver- 

100 

50 

tical,  or  two  runners  discharging 

into    a 

160 

40 

common  draft  chest. 

350 
600 
800 

30 
20 
10 

Single  or  double    discharge    runner   on 
horizontal  shaft. 

1000 

6 

Impulse  wheels. 

2000 

3 

5000 

1 

128.  Choice  of  Type  for  Low  Head. — No  definite  rules  can  be 
laid  down  for  universal  use  because  each  case  is  a  separate  prob- 
lem. Neither  is  it  possible  to  draw  any  line  between  a  high  and 
a  low  head.  AllTthat  can  be  done  is  to  assume  cases  that  are 
typical  and  establish  broad  general  conclusions.  In  any  particu- 


SELECTION  OF  TYPE  OF  TURBINE  189 

lar  case  the  engineer  can  then  decide  what  considerations  have 
weight  and  what  have  not. 

The  average  low-head  plant  has  very  little,  if  any,  storage 
capacity.  In  times  of  light  load  the  water  not  used  is  generally 
being  discharged  over  the  spillway  of  the  dam.  Economy  of 
water  on  part -load  is  thus  of  very  little  importance.  The  effi- 
ciency on  full  load  is  of  value  as  it  determines  the  amount  of  power 
that  may  be  developed  from  the  flow  available. 

Under  a  low  head  the  r.p.m.  is  normally  low  and  it  is  desirable 
to  have  a  runner  with  a  small  diameter  and  a  high  value  of  <£,  in 
order  to  secure  a  reasonable  speed.  A  high  speed  means  a 
cheaper  generator  and,  to  some  extent,  a  cheaper  turbine.  These 
were  the  factors  that  brought  about  the  development  of  the  high- 
speed turbine. 

A  low-head  plant  is  also  usually  subjected  to  a  relatively  large 
variation  in  the  head  under  which  it  operates.  When  the  head 
falls  below  its  normal  value  the  overgate  feature  of  the  high- 
speed turbine,  enabling  it  to  hold  up  the  power,  to  some  extent, 
at  a  good  efficiency,  is  a  very  valuable  characteristic. 

The  only  disadvantage-  of  the  high-speed  turbine  for  the  typical 
low-head  plant  is  that  its  maximum  efficiency  under  normal  head 
is  not  as  good  as  that  of  the  lower  speed  turbines.  However,  the 
other  advantages  outweigh  this  so  that  it  is  undoubtedly  the  best 
for  the  purpose. 

129.  Choice  of  Type  for  Medium  Head. — With  a  somewhat 
higher  head  a  limited  amount  of  storage  capacity  usually  becomes 
available  and  thus  the  efficiency  on  part-load  becomes  of  interest 
as  well  as  the  efficiency  on  full-load.     The  r.p.m.  also  approaches 
a  more  desirable  value  so  that  the  necessity  for  a  high-speed  run- 
ner disappears.     The  variation  in  head  will  generally  be  less 
serious  also,  so  that  the  overgate  feature  of  the  high-speed  turbine 
becomes  of  less  value.     The  high  efficiency  of  the  medium-speed 
turbine  fits  it  for  this  case.     The  high-speed  turbine  should  not 
be  used  unless  the  interest  on  the  money  saved  is  more  'than  the 
value  of  the  power  lost  through  the  lower  efficiency. 

130.  Choice  of  Type  for  High  Head.— For  high  heads  the 
possibility  of  extensive  storage  increases  and  the  average  oper- 
ating efficiency  then  becomes  of  more  interest  than  the  maximum 
efficiency,  especially  if  the  turbine  is  to  run  under  a  variable  load. 
Since  the  normal  speed  under  such  a  head  is  high,  a  runner  with  a 
large  diameter  and  a  low  value  of  0  may  be  desirable,  as  it  keeps 


190  HYDRAULIC  TURBINES 

the  r.p.m.  down  to  a  reasonable  limit.  The  choice  lies  between  a 
medium-speed  turbine,  a  low-speed  turbine,  or  a  tangential 
water  wheel. 

If  the  wheel  is  to  run  on  full-load  most  of  the  time,  the  high 
full-load  efficiency  of  the  medium-speed  turbine  fits  it  for  the 
place.  If  the  load  is  apt  to  vary  over  a  wide  range  and  be  very 
light  a  considerable  portion  of  the  time,  the  comparatively  flat 
efficiency  curve  of  the  tangential  water  wheel  renders  it  suitable. 
There  is  little  difference  between  the  characteristics  of  the  low 
and  medium-speed  wheels.  The  choice  between  them  is  largely 
a  matter  of  the  r.p.m.  desired,  although  there  is  some  slight 
difference  in  efficiency. 

131.  Choice  of  Type  for  Very  High  Head. — Within  certain 
limits  there  is  a  choice  between  the  low-speed  reaction  turbine 
and  the  tangential  water  wheel.  The  former  might  be  chosen 
in  some  cases  because  of  its  higher  speed  with  a  consequently 
cheaper  generator  and  the  smaller  floor  space  occupied  by  the 
unit.  The  latter  has  the  advantage  of  greater  freedom  from 
breakdowns  and  the  greater  ease  with  which  repairs  may  be 
made.  This  consideration  is  of  more  value  with  the  average 
high-head  plant  than  with  the  average  low-head  plant,  since  the 
former  is  usually  found  in  a  mountainous  region  where  it  is 
comparatively  inaccessible,  and  is  away  from  shops  where  ma- 
chine work  can  be  readily  done. 

For  extremely  high  heads  there  is  no  choice.  The  structural 
features  necessary  are  such  that  the  tangential  water  wheel  is 
the  only  type  possible.  Also  the  relatively  low  speed  of  the 
tangential  water  wheel  is  of  advantage  where  the  speed  is  in- 
herently high. 

132.  QUESTIONS  AND  PROBLEMS 

1.  For  a  given  head  and  stream  flow  available  at  a  certain  power  plant, 
what  quantities  may  be  changed  so  as  to  permit  the  use  of  various  types  of 
turbines?     Which  type  of  turbine  will  give  the  smallest  number  of  units 
in  the  plant?     Which  type  will  run  at  the  lowest  r.p.m.? 

2.  How  do  impulse  wheels  and  reaction  turbines  compare  as  to  the  maxi- 
mum efficiency  attained  by  each?     How  does  the  efficiency  of  an  impulse 
wheel  vary  with  its  size?     Why?     How  does  that  of  a  reaction  turbine 
vary  with  its  size?     Why? 

3.  For  the  same  power  under  the  same  head  compare  impulse  wheels  and 
reaction  turbines  with  respect  to  efficiency,  rotative  speed,  space  occupied, 
freedom  from  breakdown,  ease  of  repairs,  and  durability  with  silt  laden 
water. 


SELECTION  OF  TYPE  OF  TURBINE  191 

4.  How  does  the  maximum  efficiency  of  a  reaction  turbine  vary  with  the 
type  of  turbine?  For  what  type  is  it  the  highest?  Why?  For  what  type 
is  it  the  lowest?  Why? 

6.  What  are  the  disadvantages  of  a  very  low  specific  speed  reaction  tur- 
bine? What  are  its  advantages? 

6.  How  does  the  efficiency  of  the  Pelton  wheel  vary  with  its  specific  speed? 
Why? 

7.  What  is  meant  by  full-load?     What  affects  the  efficiency  of  a  tangential 
water  wheel  on  part-load? 

8.  What  affects  the  efficiency  of  a  reaction  turbine  on  part-load?     Is  the 
part-load  efficiency  a  function  of  specific  speed? 

9.  What  is  meant  by  full-gate?     By  overgate?     What  types  of  turbines 
are  overgated? 

10.  What  is  the  difference  in  the  characteristics  of  low  and  high  specific 
speed  reaction  turbines  when  run  at  the  same  speed  under  a  head  less  than 
normal?     Why? 

11.  What  are  the  advantages  and  disadvantages  of  very  high  specific 
speed  turbine  runners? 

12.  What  types  of  turbines  could  be  used  under  a  head  of  20  ft.?     Under 
200ft.?     Under  1000ft.? 

13.  What  are  the  advantages  of  a  high-speed  runner  under  very  low 
heads?     What  are  the  advantages  of  a  medium  speed  runner  under  the 
same  conditions? 

14.  What  are  the  especial  merits  of  tangential  water  wheels  for  very  high 
heads?     What  are  the  disadvantages  of  a  low-speed  reaction  turbine  for 
the  same  conditions? 

15.  The  turbine  runner  for  which  the  curves  in  Fig.  107  were  plotted  was 
23  in.  in  diameter  and  had  a  specific  speed  of  93.     The  specific  speed  of  the 
runner  for  which  the  curves  of  Fig.  117  were  drawn  was  27  and  the  diameter 
was  57  in.     Suppose  a  turbine  was  required  to  deliver  1200  h.p.  at  full-gate 
under  a  head  of  25  ft.,  find  the  size  and  r.p.m.  for  a  runner  of  each  of  these 
types.  Ans.     47.8  in.,  150  in.,  144  r.p.m.,  43.5  r.p.m. 

16.  If  the  speeds  remain  as  in  problem  (15)  while  the  head  decreases  from 
25  ft.  to  16  ft.,  find  the  power  of  each  turbine.     Ans.     648  h.p.,  465  h.p. 

17.  The   average   flow  of  a  stream  is  3000  cu.  ft.  per  second  and  the 
pondage  is  very  limited.     The  normal  head  is  30  ft.  but  is  at  times  as  low 
as  18  ft.     What  type  of  turbine  should  be  employed,  how  many  units  should 
there  be,  and  at  what  speed  will  they  run? 

Ans.     4  units  at  124  r.p.m.  probably  best. 

18.  The  average  flow  of  a  stream  is  3000  cu.  ft.  per  second.     The  normal 
head  is  30  ft.  which  is  decreased  somewhat  in  times  of  flood.     The  stream 
flow  is  fluctuating  with  long  low  water  periods,  but  there  is  considerable 
storage.     The  load  on  the  plant  also  varies  considerably.     What  type  of 
turbine  should  be  used,  how  many  units  should  there  be,  and  at  what  speed 
should  they  run? 

19.  A  turbine  is  required  to  carry  a  constant  load  of  800  h.p.  under  a 
head  of  120  ft.     There  is  considerable  storage  capacity  and  the  stream  has 
periods  of  ^low  run-off.     The  wheel  is  to  drive  a  60-cycle  alternator.     What 
type  of  turbine  should  be  used  and  what  will  be  its  speed? 


CHAPTER  XIV 
COST  OF  TURBINES  AND  WATER  POWER 

133.  General  Considerations. — Since  there  are  so  many  factors 
involved,  it  is  rather  difficult  to  establish  definite  laws  by  which 
the  cost  of  a  turbine  may  be  accurately  predicted.  No  attempt 
to  do  so  will  be  made  here,  but  a  discussion  of  the  factors  involved 
and  their  affects  will  be  given  and  the  general  range  of  prices 
stated.  A  few  actual  cases  are  cited  as  illustrations. 

A  stock  turbine  will  cost  much  less  than  one  that  is  built  to 
order  to  fulfil  certain  specifications.  This  fact  is  illustrated  by 
the  comparison  of  two  wheels  of  about  the  same  size  and  speed. 
The  specifications  of  the  stock  turbine  were  as  follows:  550  h.p. 
at  600  r.p.m.  under  a  head  of  134  ft.,  26-in.  double  discharge 
bronze  runner,  cast  steel  wicket  gates,  cast-iron  split  globe  casing 
5  ft.  in  diameter,  and  riveted  steel  draft  tube.  Weight  about 
11,500  Ib.  Price  $1750.  The  special  turbine  was  as  follows: 
500  h.p.  at  514  r.p.m.  under  a  head  of  138  ft.,  bronze  runner, 
spiral  case,  riveted  steel  draft  tube,  connections  to  header,  relief 
valve,  and  vertical  type  5000  ft.-lb.  Lombard  governor.  Price 
$4000.  The  latter  includes  a  governor,  relief  valve,  and  some 
connections  which  the  former  did  not,  but  the  difference  in  cost 
is  more  than  the  price  of  these. 

The  cost  of  the  turbine  is  also  affected  by  the  quality  and 
quantity  of  material  entering  into  it,  the  grade  of  workmanship, 
and  the  general  excellence  of  the  design.  With  the  $4000  tur- 
bine cited  in  the  preceding  paragraph  another  may  be  compared 
which  is  of  superior  design.  The  specifications  for  the  latter 
were  as  follows:  550  h.p.  at  600  r.p.m.  under  142-ft.  head,  single 
discharge  bronze  runner,  spiral  case  with  30-in.  intake,  cast  steel 
wicket  gates,  bronze  bushed  guide  vane  bearings,  riveted  steel 
draft  tube,  lignum  vitae  thrust  bearing,  oil  pressure  governor 
sensitive  to  0.5  per  cent.  The  guaranteed  efficiencies  were 

83  per  cent,  at  410  h.p. 

84  per  cent,  at  500  h.p. 
83  per  cent,  at  550  h.p. 

192 


COST  OF  TURBINES  AND  WATER  POWER      193 

(Nothing  was  said  about  efficiency  in  the  preceding  case.) 
Weight  of  turbine  30,000  lb.,  of  governor  3000  Ib.  Price  $6000. 
The  turbine  just  quoted  was  similar  to  one  previously  built 
and  the  patterns  required  only  slight  modification.  Where  an 
entirely  new  design  is  called  for  the  cost  will  be  greater  still,  as 
is  evidenced  by  the  bid  of  another  firm,  as  follows:  550  h.p.  at 
600  r.p.m.  under  142-ft.  head,  single  discharge  cast  iron  runner, 
spiral  case,  cast  steel  guide  vanes,  cast  steel  flywheel,  oil  pres- 
sure governor,  connections  to  header,  30-in.  hand-operated  gate 
valve,  riveted  steel  draft  tube,  and  relief  valve.  The  guaranteed 
efficiencies  were 


Per  cent,  of  max.  h.p. 

81.5  per  cent,  at  

100 

84.5  per  cent,  at     . 

90 

84.5  per  cent  at 

85 

82.5  per  cent,  at  

75 

79.5  per  cent,  at  .   . 

60 

Weight  of  turbine  complete  38,000  lb.  Price  $8740.  This  last 
turbine  includes  a  few  items  that  the  former  does  not,  but  the 
difference  in  cost  cannot  be  accounted  for  by  them.  It  will  be 
noted  that  a  flywheel  was  deemed  necessary  here,  while  it  was 
not  used  on  any  of  the  others.  Compare  the  weights  and  costs 
of  these  last  two  turbines  with  the  weight  and  cost  of  the  stock 
turbine  first  mentioned. 

134.  Cost  of  Turbines. — The  cost  of  a  turbine  depends  upon 
its  size  and  not  upon  its  power.  Since  the  power  varies  with 
the  head,  it  is  apparent  that  the  cost  per  h.p.  is  less  as  the  head 
increases.  Thus  a  certain  16-in.  turbine  (weight  =  7000  lb.) 
without  governor  or  any  connections  may  be  had  for  $1000. 
Under  various  heads  the  cost  per  horsepower  would  be  as 
follows : 


Head 

H.p. 

[  Cost  per  h.p. 

30ft  

52 

$19  20 

60  ft 

148 

6  75 

100  ft 

318 

3  14 

194  HYDRAULIC  TURBINES 

One  would  not  be  warranted  in  saying,  however,  that  under 
10-ft.  head  a  turbine  would  cost  $100  per  horsepower  because 
the  above  would  develop  only  10  h.p.  under  that  head.  Neither 
would  one  be  justified  in  saying  that,  since  this  turbine  would 
develop  1650  h.p.  under  300-ft.  head,  that  the  cost  per  horse- 
power might  be  only  $0.605.  Under  a  10-ft.  head  a  much 
lighter  and  cheaper  construction  would  be  entirely  reasonable, 
while  under  a  300-ft.  head  the  turbine  would  have  to  be  built 
stronger  and  better  than  this  one  was. 

For  a  given  head,  the  greater  the  power  of  the  turbine  the  less 
the  cost  per  horsepower  will  be.  Also  for  a  given  head  and 
power,  the  higher  the  speed,  the  smaller  the  wheel,  and  conse- 
quently the  less  the  cost.  Compare  the  600-r.p.m.  reaction 
turbines  in  Art.  133  with  the  following,  which  is  a  double  over- 
hung tangential  water  wheel  at  120  r.p.m.  The  horsepower 
is  500  under  134-ft.  head.  Oil  pressure  governor  is  included,  but 
no  connections  to  penstock  are  furnished.  Weight  80,000  Ib. 
Price  $8900. 

These  last  differences  are  very  much  magnified  if  we  combine 
the  cost  of  the  generator  with  that  of  the  turbine.  The  follow- 
ing are  some  generator  quotations.  The  first  is  that  of  a  gener- 
ator at  a  special  speed.  The  second  is  that  of  a  generator  of 
somewhat  better  construction  than  the  first  but  of  a  standard 
speed.  The  others  are  all  standard  speeds. 

150  kv.-a.,  2400  volts,  3-phase,  60-cycle,  124  r.p.m.  $4850. 

150  kv.-a.,  2400  volts,  3-phase,  60-cycle,  120  r.p.m.  $3300. 

(Weight  17,210  Ib.) 

300  kv.-a.,  2400  volts,  3-phase,  60-cycle,  120  r.p.m.  $4700. 

(Weight  25,520  Ib.) 

350  kv.-a.,  2400  volts,  3-phase,  60-cycle,  514  r.p.m.  $2330. 

350  kv.-a.,  2400  volts,  3-phase,  60-cycle,  600  r.p.m.  $2100. 

Taking  the  highest  priced  600-r.p.m.. turbine  and  combining  it 
with  the  350-kv.-a.  generator  we  get  a  total  of  $10,850.  Adding 
the  cost  of  the  120-r.p.m.  turbine  to  that  of  the  300-kv.-a.  genera- 
tor we  get  a  total  of  $13,600  for  a  smaller  amount  of  power. 

Prof.  F.  J.  Seery  has  derived  the  following  empirical  formula 
based  upon  the  list  prices  of  35  wheels  made  by  20  manufacturers. 

Log  X  =  A  +  D/B,  in  which  X  is  the  cost  in  dollars  for  a 
single  stock  runner  with  gates  and  crown  plates  suitable  for 


COST  OF  TURBINES  AND  WATER  POWER      195 

setting  in  a  flume.  The  value  of  A  ranges  from  1.09  to  2.17, 
but  the  usual  value  is  about  1.9.  The  value  of  B  varies  from 
40  to  83  with  a  usual  value  of  about  50.  These  prices  are  sub- 
ject to  discounts  also.  The  cost  of  a  draft  chest  for  a  twin  run- 
ner will  be  given  by 

X  =  0.045  D2-25,  in  which  X  is  in  dollars  and  D  is  the 
diameter  of  the  runners  in  inches. 

The  cost  of  the  casing  increases  these  values  very  greatly, 
as  some  spiral  cases  may  cost  much  more  than  the  runner.  A 
single  case  may  be  cited  of  a  pair  of  20-in.  stock  runners  in  a 
cylinder  case  with  about  30  ft.  of  5  ft.  steel  penstock.  Each 
runner  discharges  into  a  separate  draft  tube  about  3  ft.  long. 
The  power  is  150  h.p.  under  30-ft.  head.  The  cost  was  $2000. 

A  few  quotations  are  here  given.  A  reaction  turbine  to  de- 
velop 4000  h.p.  at  600  r.p.m.  unc(er  375  ft.  head  and  weighing 
90,000  Ib.  would  cost  $14,000.  Another  reaction  turbine  of 
10,000  h.p.  under  565-ft.  head  cost  $37,000.  In  the  latter  case 
the  governor,  pressure  regulator,  and  the  generator  were  included. 
The  building,  crane,  transformer  room,  etc.,  cost  $20,000  for 
this  installation.  A  tangential  water  wheel  of  2500  h.p.  under 
1200-ft.  head  cost  $12,000,  while  another  of  4500  h.p.  under 
1700-ft.  head  cost  $8,000. 

As  has  been  stated,  the  cost  of  a  turbine  varies  between  fairly 
wide  limits  due  to  difference  in  design,  workmanship,  and  com- 
mercial conditions.  The  cost  per  h.p.  is  also  less  the  higher  the 
head  or  the  greater  the  power.  In  a  general  way  it  can  be  said  to 
vary  between  $2  and  $30  per  horse-power  and  according  to  the 
following  table: 


Head 

Cost  per  h.p. 

Cost  of  building  per  h.p. 

60  ft  

$30-$7 

$30-$4 

100-600  ft 

$12-$2 

$  7-$2 

500-2000  ft  

$  8-$2 

$  7-$2 

The  cost  of  the  turbine  is  usually  only  about  6  per  cent,  of 
the  total  cost  of  the  power  plant.  It  scarcely  pays,  therefore, 
to  buy  a  cheap  turbine  when  the  money  saved  is  such  a  small 
portion  of  the  entire  investment. 

135.  Capital  Cost  of  Water  Power. — The  capital  cost  of  water 
power  includes  the  investment  in  land,  water  rights,  storage 


196 


HYDRAULIC  TURBINES 


reservoirs,  dams,  head  races  or  canals,  pipe  lines,  tail  race, 
power  house,  equipment,  transmission  lines,  interest  on  money 
tied  up  before  plant  can  be  put  into  operation,  and  often  the  cost 
of  an  auxiliary  power  or  heating  plant. 

The  capital  cost  per  horsepower  is  less  as  the  capacity  of 
the  plant  is  greater.  This  is  shown  by  the  following  table  from 
the  report  of  the  Hydro-Electric  Power  Commission  of  the 
Province  of  Ontario.  The  proposed  plant  was  to  be  located  at 
Niagara  Falls. 

TABLE  9 


Items 

50,000  h.p. 

100,000  h.p. 

Tunixel  tail  race 

$1,250,000 

$1  250  000 

Headworks  and  canal  
Wheel  pit  .          

450,000 
500,000 

450,000 
700,000 

Power  house 

300,000 

600  000 

Hydraulic  equipment  

1,080,000 

1,980,000 

Electric  equipment  

760,000 

1,400,000 

Transformer-  station  and  equipment  
Office  building  and  machine  shop  
Miscellaneous           

350,000 
100,000 
75,000 

700,000 
100,000 
75,000 

Engineering  etc     10  per  cent 

$4,865,000 
485  000 

$7,255,000 
725  000 

Interest  2  years  at  4  per  cent 

$5,350,000 
436,560 

$7,980,000 
651,168 

Total  capital  cost    

$5,786,560 

$8,631,168 

Capital  cost  per  horsepower 

$114 

$86 

The  cost  per  unit  capacity  is  usually  less  as  the  head  increases. 
This  is  illustrated  by  the  following  table  taken  from  Mead's 
"  Water  Power  Engineering." 


Capital  cost  per  h.p. 

Capacity 
horsepower 

Head 

Without  dam 

With 
dam 

With  dam 
and  electrical 
equipment 

With  dam,  electric 
equipment,  and 
transmission  line 

8000 

18 

$63.50 

86 

115 

150 

8000 

80 

21.00 

39 

60 

90 

COST  OF  TURBINES  AND  WATER  POWER       197 


The  capital  cost  may  range  from  $40  to  $200  per  horsepower, 
but  the  average  value  is  about  $100. l 

136.  Annual  Cost  of  Water  Power. — The  annual  cost  of  water 
power  will  be  the  sum  of  the  fixed  charges  and  the  operating 
expenses.  The  former  will  cover  interest  on  the  capital  cost, 
taxes,  insurance,  depreciation,  and  any  other  items  that  are  con- 
stant. The  latter  includes  repairs,  supplies,  labor,  and  any 
other  items  that  vary  according  to  the  load  the  plant  carries. 
The  annual  cost  per  horsepower  is  the  total  annual  cost  divided 
by  the  horsepower  capacity  of  the  plant. 

The  total  annual  cost  will  vary  with  the  number  of  hours  the 
plant  is  in  service  and  also  with  the  load  carried.  The  cost  will 
be  a  maximum  when  the  plant  carries  full  load  24  hours  per  day 
and  365  days  per  year.  It  will  be  a  minimum  when  the  plant  is 
shut  down  the  entire  year,  being  then  only  the  fixed  charges.  (See 


Hours  per  Year 
FlG.   119. 


8760 


Fig.  119).  It  is  evident  that  the  annual  cost  per  horsepower 
depends  upon  the  conditions  of  operation. 

However,  under  the  usual  conditions  of  operation,  the  annual 
cost  may  be  said  to  vary  from  $10  to  $30  per  horsepower. 

137.  Cost  of  Power  per  Horsepower-hour. — In  order  to 
have  a.  true  value  of  the  cost  of  power  it  is  necessary  to  consider 
both  the  load  carried  and  the  duration  of  the  load.  While  the 
annual  cost  per  horsepower  will  be  a  maximum  when  the  plant 
carries  full  load  continuously  throughout  the  year,  the  cost  per 
horsepower-hour  will  be  a  minimum.  Thus  suppose  the  annual 
cost  per  horsepower  of  a  plant  in  continuous  operation  on  full- 
load  is  $20.  The  cost  per  horsepower-hour  is  then  0.228  cents. 
Suppose  that  the  plant  is  operated  only  12  hours  per  day  and  that 

1  For  specific  cases  see  Mead's  "Water  Power  Engineering,"  p.  650. 


198 


HYDRAULIC  TURBINES 


the  annual  cost  per  horsepower  then  becomes  $17,  the  cost  of 
power  will  be  0.388  cents  per  horsepower-hour.  So  far  the  load 
has  been  treated  as  constant;  we  shall  next  assume  that  it  varies 
continuously  and  that  it  has  a  load  factor  of  25  per  cent.  By  that 
is  meant  that  the  averagejoad  is  25  per  cent,  of  the  maximum. 
If  the  plant  be  operated  12  hours  per  day  as  before,  the  annual  cost 
per  maximum  horsepower  may  still  be  $17,  but  the  annual  cost 
per  average  horsepower  will  be  $68.  This  latter  divided  by 
4380  hours  gives  1.55  cents  per  horsepower-hour.  It  is  clear, 
then,  that  the  cost  of  power  per  horsepower-hour  depends 
very  greatly  upon  the  load  curve.  It  may  range  anywhere  from 


I 


Load  Factor 
FlG.    120. 


0.40  cents  to  1.3  cents  per  horsepower-hour  and  more  if  the  load 
factor  is  low.     (See  Fig.  120.) 

138.  Sale  of  Power. — If  power  is  to  be  sold,  one  of  the  first 
requirements  generally  is  that  the  output  of  the  plant  should  be 
continuous  and  uninterrupted.  Such  a  plant  should  possess  at 
least  one  reserve  unit  so  that  at  any  time  a  turbine  can  be  shut 
down  for  examination  or  repair.  This  adds  somewhat  to  the 
cost  of  the  plant.  The  larger  the  units  the  more  the  added  cost 
of  this  extra  unit  will  be.  On  the  other  hand  small  units  are 
undesirable  since  a  large  number  of  them  make^the  plant  too 
complicated.  Also  the  efficiency  of  the  smaller  wheels  will  be 
less  than  that  of  the  larger  sizes.  Unless  the  water-supply  is 
fairly  regular,  storage  reservoirs  will  be  necessary  and  often  auxil- 


COST  OF  TURBINES  AND  WATER  POWER       199 

iary  steam  plants  are  essential  in  order  that  the  service  may  not 
be  suspended  either  in  time  of  high  or  low  water. 

A  market  for  the  power  created  is  essential.  If  the  demand  for 
the  power  does  not  exist  at  the  time  the  plant  is  projected/  there 
should  be  very  definite  assurance  that  the  future  growth  of  in- 
dustry will  be  sufficient  to  absorb  the'output  of  the  plant. 

If  the  plant  is  to  be  a  financial  success,  the  price  at  which 
power  is  sold  should  exceed  the  cost  of  generation  by  a  reasonable 
margin  of  profit.  The  price  for  which  the  power  may  be  sold 
is  usually  fixed  by  the  cost  of  its  production  in  other  ways.  This 
point  should  be  carefully  investigated  and,  if  the  cost  from  other 
sources  is  less  than  the  cost  of  the  water  power  plus  the  profit, 
the  proposition  should  be  abandoned. 

139.  Comparison  with  Steam  Power. — It  is  necessary  to  be 
able  to  estimate  the  cost  of  other  sources  of  power  in  order  to  tell 


Load  Factor 
FIG.  121. — Comparison  of  costs  of  steam  and  water  power. 

whether  a  water-power  plant  will  pay  or  not.  Also  it  is  often 
essential  to  figure  on  the  cost  of  auxiliary  power.  As  steam  is 
the  most  common  source  of  power  and  is  typical  of  all  others,  our 
discussion  will  be  confined  to  it. 

In  general  the  capital  cost  of  a  steam  plant  is  less  than  that  of  a 
water-power  plant.  It  varies  from  $40  to  $100  per  horsepower, 
with  an  average  value  of  about  $60  per  horsepower.  Deprecia- 
tion, repairs,  and  insurance  are  at  a  somewhat  higher  rate  but, 
nevertheless,  the  fixed  charges  are  less  than  for  water  power. 

The  amount  of  labor  necessary  is  greater  and  this,  together 


200 


HYDRAULIC  TURBINES 


with  the  cost  of  fuel  and  supplies,  causes  the  operating  expenses 
to  be  higher  than  with  the  water  power.  The  ,  total  cost  of 
power  for  the  two  cases  is  compared  in  Fig.  121.  As  to  whether 
the  cost  of  steam  power  in  a  given  case  is  greater  or  less  than  that 
of  water  power  at  100  per  cent,  load  factor  it  is  impossible  to 
state  without  a  careful  investigation.  But  it  is  clear  that,  as  a 
rule,  the  cost  of  steam  power  is  less  when  the  plant  is  operated 
but  a  portion  of  the  year  or  when  the  load  factor  is  low.  Thus  a 
water-power  plant  is  of  the  most  value  when  operated  at  high 
load  factor  throughout  the  year. 

The  annual  cost  of  steam  power  per  horsepower  is  very  high 
for  small  plants  but  for  capacities  above  500  h.p.  it  does  not  vary 
so  widely.  Its  value  depends  upon  the  capacity  of  the  plant,  the 
load  factor,  and  the  length  of  time  the  plant  is  operated.  It  may 
be  anywhere  from  $20  to  $70,  though  these  are  by*jnojneans 
absolute  limits.  / 

Since  the  operating  expenses  are  of  secondary  importance  in  a 
water-power  plant,  the  annual  cost  per  horsepower  will  not  be 
radically  different  for  different  conditions  of  operation.  But 
with  a  steam  plant  the  annual  cost  per  horsepower  varies  widely 
for  different  conditions  of  operation  on  account  of  the  greater 
effect  of  the  variable  expenses.  It  is  much  better  to  reduce  all 
costs  to  cents  per  horsepower  hour.  The  accompanying  table 
gives  the  usual  values  of  the  separate  items  that  make  up  the 
cost  of  steam  power,  reduced  to  cents  per  horsepower  hour. 


Items 

Min.,  cents 

Max.,  cents 

Fuel                 

0.20 

0.75 

Supplies                                                               .  .    . 

0  03 

0.06 

Labor  ....        

0.07 

0.14 

Administration 

0  02 

0.15 

Repairs                    

0.05 

0.10 

Fixed  charges            .  .                  

0  30 

0.45 

Total  cost  per  horsepower  hour  '.  

0.67 

1.65 

The  following  comparison  is  made  by  C.  T.  Main  in  Trans. 
A.  S.  M.  E.,  Vol.  XIII,  p.  140.  The  location  was  at  Lawrence, 
Mass.  Fixed  charges  were  estimated  on  the  following  basis : 


COST  OF  TURBINES  AND  WATER  POWER      201 


Steam, 
per  cent. 

Water, 
per  cent. 

Interest     .                             .            .            

5   0 

5 

Depreciation  

3.5 

2 

Repairs  

2.0 

1 

Insurance  .  .                                    ... 

2  0 

1 

Total  

12.5 

9 

For  a  steam  plant  at  that  location  the  capital  cost  was  taken 
as  $65  per  horsepower.  The  annual  cost  per  horsepower  was  as 
follows : 

Fixed  charges  12.5  per  cent $8. 13 

Fuel 8.71 

Labor 4.16 

Supplies 0.80 

Total  annual  cost  per  horsepower $21 .80 

For  a  water  plant  the  cost  of  the  power  house  and  equipment 
was  taken  as  $65  while  the  cost  of  dams  and  canals  at  that  place 
averaged  $65  also,  making  a  total  capital  cost  of  $130  per  horse- 
power. The  annual  cost  per  horsepower  was  as  follows : 

Fixed  charges  9  per  cent . .  . , $11 . 70 

Labor  and  supplies 2 . 00 


Total  annual  cost  per  horsepower $  13 . 70 

However,  for  the  case  in  question,  a  steam-heating  plant  was 
necessary  and  its  cost  was  divided  by  the  horsepower  of  the 
plant  giving  the  capital  cost  of  the  auxiliary  steam  plant  as  $7.50 
per  horsepower  of  the  power  plant.  The  cost  of  its  operation 
based  upon  the  power  plant  would  ;be, 

Fixed  charges  at  12.5  per  cent $0 .94 

Coal 3.26 

Labor 1.23 

Total  cost  of  heating  per  horsepower  of  plant ....  $5. 43 

Adding  this  to  the  cost  of  the  power  we  obtain  the  total  cost 
of  the  water  power  to  be  $19.13  per  horsepower  per  year.  Evi- 


202  HYDRAULIC  TURBINES 

dently  the  difference  in  favor  of  the  water  power  will  be  $2.67  per 
horsepower. 

The  cost  of  any  kind  of  power  will  evidently  vary  in  different 
portions  of  the  country  and  it  is  impossible  to  lay  down  absolute 
facts  of  universal  application.  In  places  near  the  coal  fields  the 
cost  of  steam  power  will  be  a  minimum  and  it  may  be  impossible 
for  water  power  to  complete  with  it.  However  where  the  cost 
of  fuel  is  high  water  power  may  be  a  paying  proposition  even 
though  its  cost  may  be  relatively  high. 

140.  Value  of  Water  Power. — The  value  of  a  water  power  is 
somewhat  difficult  to  establish  as  it  depends  upon  the  point  of 
view.  However,  the  following  statements  seem  reasonable: 

An  undeveloped  water  power  is  worth  nothing  if  the  power, 
when  developed,  is  not  more  economical  than  steam  or  other 
power.  If  the  power,  when  developed,  can  be  produced  cheaper 
than  other  power,  then  the  value  of  the  water  rights  would  be  a 
sum  the  interest  on  which  would  equal  the  total  annual  saving 
due  to  the  use  of  the  latter.  Thus,  referring  to  the  case  of  Mr. 
Main  cited  in  the  preceding  article,  suppose  the  water  supply  is 
capable  of  developing  10,000  h.p.  The  annual  saving  then  due 
to  its  use  would  be  $26,700  as  compared  with  steam.  Its  value 
is  then  evidently  a  sum  the  interest  on  which  would  be  $26,700 
per  year. 

A  power  that  is  already  developed  must  be  considered  on  a  dif- 
ferent J)asis.  If  the  power  cannot  be  produced  cheaper  than  that 
from  any  other  available  source,  the  value  of  the  plant  is  merely 
its  first  cost  less  depreciation,  or  from  another  point  of  view  the 
sum  which  would  erect  another  plant,  such  as  a  steam  power 
plant,  of  equal  capacity.  If  the  water  power  can  be  produced 
cheaper  than  any  other,  the  value  of  the  plant  will  be  its  first 
cost  less  depreciation  added  to  the  value  of  the  water  right  as 
given  in  the  preceding  paragraph. 

141.  QUESTIONS  AND  PROBLEMS 

1.  What  are  the  general  factors  that  affect  the  cost  of  a  turbine  of  a 
given  speed  and  power.  ? 

2.  What  factors  affect  the  cost  of  a  turbine  per  h.p.? 

3.  What  is  meant  by  capital  cost  of  water  power?     What  items  does  it 
include?     How  is  this  cost  per  h.p.  affected  by  the  head  and  by  the  size  of 
the  plant? 

4.  What  is  meant  by  the  annual  cost  of  water  power?     How  is  it  com- 


COST  OF  TURBINES  AND  WATER  POWER      203 

puted?     When  will  it  be  a  maximum  and  when  a  minimum  for  a  given 
plant? 

5.  How  is  cost  per  h.p.  hour  computed?     Upon  what  factors  does  it 
depend?     How  does  it  vary  as  a  function  of  load  factor?     When  is  it  a 
maximum  for  a  given  plant  and  when  is  it  a  minimum?     What  can  its 
maximum  value  be? 

6.  How  do  water  and  steam  power  compare  in  general  as  to  capital  cost 
per  h.p.  and  hence-  as  to  fixed  charges?     How  do  they  compare  as  to  oper- 
ating expenses.     How  do  the  total  annual  costs  and  the  cost  per  h.p.  hour 
vary  for  each  as  functions  of  load  factor? 

7.  How  is  it  to  be  determined  beforehand  whether  a  water  power  plant 
will  pay  or  not? 

8.  How  is  the  value  of  a  water  right  to  be  determined? 

9.  How  is  the  value  of  an  existing  water  power  plant  to  be  computed? 
Can  there  be  any  doubt  about  the  correctness  of  the  method? 

10.  Suppose  you  were  called  upon  to  make  a  report  upon  a  water-power 
development,  the  only  information  given  being  the  head  available  and  the 
location  for  the  plant,  together  with  an  assurance  of  a  market  for  all  power 
produced.     How  would  you  determine:  (a)  Amount  of  power  that  can  be 
developed;  (6)  How  much  storage  capacity  should  be  provided;  (c)  Whether 
the  plant  should  be  built  at  all;  (d)  Value  of  the  water  right;  (e)  Size  of 
penstock  to  be  used;  (/)  Type  of  turbine  to  be  used;  (g)  Number,  size  and 
speeds  of  units  to  be  used? 

11.  If  steam  power  costs  $20  per  h.p.  per  year  and  water  power  can  be 
produced  for  $19  per  h.p.  per  year,  what  would  be  the  value  of  an  undevel- 
oped water  right  of  5000  h.p? 

12.  A  water  power  plant  cost  $100  per  h.p.  and  is  estimated  to  have  de- 
preciated 15  per  cent.     If  it  costs  $20  per  h.p.  per  year  to  produce  power 
from  it  in  a  place  where  steam  power  would  cost  $23  per  h.p.  per  year,  what 
is  the  value  of  the  development? 


CHAPTER  XV 
DESIGN  OF  THE  TANGENTIAL  WATER  WHEEL 

142.  General  Dimensions.1  —  Assume  that  the  head,  speed,  and 
power  for  a  proposed  water  wheel  are  known,  these  values  being 
so  selected  as  to  give  the  specific  speed  necessary  for  the  type  of 
impulse  wheel  desired.  It  is  to  be  understood  that  the  head  is 
that  at  the  base  of  the  nozzle,  and  the  power  is  the  output  corre- 
sponding to  one  jet.  The  velocity  of  the  jet  is  given  by  the 
equation,  V\  =  cv\/2gh,  where  the  value  of  the  velocity  coeffi- 
cient may  be  taken  as  0.98.  (See  Fig.  89,  page  114.)  Since 
B.h.p.  =  qhe/8.8,  we  may  write 

8.8  X  B.h.p.  wd2  — 

-~ 


where  d  is  the  diameter  of  the  jet  in  inches.     From  this  the  value 
of  d  may  be  found  to  be 


The  diameter  of  the  wheel  may  be  found  from  equation  (47), 
which  gives 

1840  ^Vh 

•  v  ;  N 

where  D  is  the  diameter  in  inches  of  the  "  impulse  circle,"  which 
is  the  circle  tangent  to  the  center  line  of  the  jet.  The  overall 
diameter  of  the  runner  depends  upon  the  dimensions  of  the  buck- 
ets. The  value  of  <f>e  is  from  0.43  to  0.47. 

143.  Nozzle  Design.  —  The  nozzle  tip  and  needle  should  be  so 
proportioned  as  to  give  a  constantly  decreasing  stream  area  from 
a  point  within  the  nozzle  to  a  point  in  the  jet  beyond  the  tip  of 
the  needle,  so  that  the  water  may  be  continuously  accelerated. 
This  must  be  so  for  every  position  of  the  needle.  The  curve  of 

1  In  this  book  only  the  hydraulic  features  of  design  will  be  considered. 
No  space  will  be  devoted  to  the  determination  of  dimensions  which  can  be 
computed  by  the  usual  methods  of  machine  design. 

204 


DESIGN  OF  THE  TANGENTIAL  WATER  WHEEL       205 

the  needle  must  therefore  change  from  convex  to  concave  and 
the  point  of  inflection  must  be  at  a  diameter  greater  than  that 
of  the  nozzle'tip,  otherwise  the  water  will  tend  to  leave  the  needle 
at  the  smaller  openings  with  a  resulting  tendency  to  corrosion. 
It  is  also  desirable,  for  the  sake  of  the  governor  action,  that  the 
rate  of  discharge  vary  approximately  in  direct  proportion  to  the 
linear  movement  of  the  needle.  (See  Fig.  89.) 

The  diameter  of  the  orifice  of  the  nozzle  tip  must  be  greater 
than  the  diameter  of  the  jet,  due  to  the  contraction  of  the  latter 
and  also  to  the  space  taken  up  by  the  needle  tip,  which  is  never 
entirely  withdrawn.  At  wide  open  setting  the  needle  tip  may 


FIG.  122. 

occupy  10  per  cent,  or  more  of  the  area  of  the  orifice.  The  re- 
maining area,  through  which  the  water  passes,  may  be  computed 
from  the  area  of  the  jet  by  the  use  of  a  coefficient  of  contraction, 
typical  values  for  which  are  given  in  Fig.  89, .page  114.  In 
reality  the  effective  area  of  the  nozzle  is  that  perpendicular  to 
the  stream  lines  and  is  the  surface  of  the  frustum  of  a  cone,  whose 
elements  are  perpendiculars  dropped  from  the  edge  of  the  orifice 
to  the  needle.  This  area  is  slightly  greater  than  that  in  the  plane 
of  the  orifice.  The  nozzle  tip  diameter  shoud  be  computed  for 
a  size  of  jet  sufficiently  large  to  carry  the  maximum  load  on  the 
wheel. 

144.  Pitch  of  Buckets. — In  Fig.  122  the  bucket  A  has  just  com- 
pletely intercepted  the  jet.  If  the  particle  of  water  at  C  is  to 
hit  bucket  B  it  must  do  so  before  the  latter  reaches  point  E. 


206 


HYDRAULIC  TURBINES 


The  time  for  a  particle  of  water  to  go  from  C  to  E  is  t  =  l/V\. 
If  in  the  same  time  bucket  B  reaches  E,  we  have,  t  =  y/u. 
But  co  =  UQ/TQ,  and  equating  the  two  values  of  t,  we  obtain 
7  =  (Wro)  (l/Vi)  =  (UQ/VI)  (//r0).  Since  l/r0  =  2  sin5  and 
0  =  26  —  r,  we  have 


=  26  -  2  Y  sin  8 


(62) 


But  this  value  of  the  pitch  angle  would  be  such  as  to  permit  the 
particle  of  water  to  merely  touch  the  bucket  before  the  latter 
swung  up  out  of  its  line  of  action.  In  order  to  permit  the  water 
to  flow  over  the  bucket  a  closer  spacing  than  this  is  required.  The 
time  necessary  for  flow  over  the  bucket  may  be  represented  by 
t'  =  l'/vf,  where  V  is  the  length  of  path  and  vf  the  velocity 
relative  to  the  bucket,  a  mean  value  being  chosen  between  Vi 


FIG.  123. 

and  t;2.  It  appears  rather  difficult  to  express  this  readily  in  a 
simple  formula  and  the  practical  procedure  appears  to  be  to 
assume  an  approximate  spacing  for  the  buckets  and  then  compute 
the  probable  time  required  for  a  particle  of  water  to  complete 
its  flow.  As  a  preliminary  trial  value  we  may  assume  the  above 
value  to  be  reduced  by  20  per  cent.,  in  which  case  the  number  of 
buckets  n  may  be  found  by 

n  =  27T/0.8  X  0  (63) 

As  noted  above,  this  value  should  be  checked  by  a  numerical 
computation. 

So  far  the  bucket  has  been  considered  as  if  all  points  on  its  lower 
edge  were  at  the  same  distance  from  the  axis,  whereas  the  buckets 
of  the  present  day  have  some  form  of  notch  in  this  edge,  as  may 
be  seen  in  Fig.  123.  The  water  which  strikes  the  bucket  during 


DESIGN  OF  THE  TANGENTIAL  WATER  WHEEL       207 

\ 

the  early  part  of  its  course  flows  out  along  the  arc  ST,  while  that 
which  strikes  during  the  latter  part,  when  the  bucket  is  quite 
inclined  to  the  axis  of  the  jet,  flows  out  across  the  portion  X Y. 
The  effect  of  this  prolongation  of  the  bucket  beyond  the  part  MN 
is  to  permit  the  water  which  strikes  it  just  before  it  reaches  point 
E  in  Fig.  122  to  fully  act.  In  other  words,  referring  to  Fig.  122, 
the  entrance  edge  of  the  bucket  describes  the  arc  CE,  but  the 
extreme  discharge  edge  describes  a  larger  arc.  This  permits  the 
use  of  fewer  buckets  on  a  wheel  without  involving  any  loss.  It  is 
not  desirable  to  extend  the  part  MN  to  the  same  radius  because 
that  would  make  conditions  less  favorable  when  the  bucket  first 
enters  the  jet. 

For  a  high  specific  speed  wheel  the  runner  diameter  becomes 
relatively  smaller  for  the  same  jet  diameter  and  this  shortens 
the  length  of  the  path  CE  (Fig.  122).  In  order  that  all  the  water 
may  be  fully  utilized  it  is  necessary  to  reduce  the  time  required 
for  the  particle  of  water  at  C  to  catch  up  with  bucket  B.  This 
can  be  done  by  reducing  the  pitch. 

But  there  is  evidently  a  limit  to  this  for  mechanical  reasons, 
as  a  certain  amount  of  metal  is  necessary  in  order  that  each  bucket 
may  be  securely  fastened  to  the  rim.  Furthermore,  the  closer  the 
buckets  are  placed  together  the  quicker  must  the  water  dis- 
charged be  gotten  out  of  the  way  of  the  following  bucket.  This 
means  that  it  must  leave  with  a  higher  residual  velocity,  which 
means  in  turn  that  the  kinetic  energy  lost  af;  discharge  is  greater. 
This  is  one  reason  why  the  efficiency  of  an  impulse  wheel  is 
less,  if  the  specific  speed  is  too  high.  After  the  number  of  buckets 
on  a  given  wheel  has  been  made  a  maximum,  the  only  other  means 
of  increasing  the  specific  speed  is  to  lengthen  the  buckets  still 
more,  but  this  evidently  soon  reaches  its  limit. 

The  curve  representing  the  end  of  the  portion  of  the  jet  in- 
tercepted by  the  bucket  A  may  be  drawn  by  plotting  the  path 
of  the  tip  of  the  bucket  relative  to  the  jet.  By  computing  the 
time  necessay  for  bucket  B  to  get  to  its  extreme  right  hand  posi- 
tion and  then  by  moving  CF  the  distance  the  water  would  travel 
in  the  same  time  interval,  it  is  apparent  whether  any  water 
is  not  utilized  or  not,  and  also  the  amount  wasted  can  be  approxi- 
mately determined.1 

'See,  "Theory  of  the  Tangential  Waterwheel,"  by  R.  L.  Daugherty  in 
Cornell  Civil  Engineer,  Vol.  22,  p.  164  (1914). 


208  HYDRAULIC  TURBINES 

145.  Design  of  Buckets. — The  general  dimensions  of  the 
buckets  must  bear  some  relation  to  the  size  of  the- jet  and  experi- 
ence shows  that  the  width  of  the  bucket  should  be  at  least  three 
times  that  of  jet  and  the  length  about  the  same  or  a  little  more.1 
The  exact  dimensions  should  be  determined  for  individual  cases 
and  naturally  vary  somewhat  with  the  specific  speed  of  the  wheel. 

In  order  to  get  the  proper  bucket  shape,  curves  may  be  plotted 
showing  the  path  of  the  jet  relative  to  the  wheel,  as  in  Fig.  124. 
In  the  figure  only  one  such  curve  is  shown,  that  for  the  top  of  the 
jet,  and  also  we  consider  here  only  one  section,  that  in  the  plane 
of  the  paper;  but  other  stream  lines  and  other  parallel  planes 
should  also  be  used.  It  should  be  noted  that  this  is  the  relative 
path  for  the  free  jet  only.  As  soon  as  the  water  flows  over  the 
buckets  its  absolute  velocity  is  altered,  and  consequently  its 


\ 
Path  of  Point  on  Wheel 

Eelative  to  Jet 

FIG.  124. 

relative  velocity  and  path  are  different.  But  as  the  function  of 
these  curves  is  merely  to  aid  in  determining  the  entrance  condi- 
tions they  are  sufficient.  As  the  bucket  first  enters  the  jet,  the 
water  flows  in  over  the  lip  in  the  center  of  the  notch,  as  MN  in 
Fig.  123.  It  is  only  after  the  bucket  has  travelled  somewhat 
farther  that  the  water  strikes  it  fully  on  the  "-splitter."  As  seen 
in  Fig.  124,  the  face  of  the  bucket  along  the  lip  should  be  such 
that  the  surface  is  approximately  tangent  to  the  relative  path 
of  the  water,  in  order,  not  to  have  any  loss  of  energy  at  this 
point.  After  the  bucket  has  moved  along  to  another  position 
where  the  jet  strikes  it  in  another  place,  the  shape  of  that  portion 
will  be  determined  in  the  same  way,  but  of  course  another  por- 
tion of  the  curve  will  be  used.  Also  the  "splitter"  should  be 
approximately  perpendicular  to  the  relative  path.  (See  Fig.  23.) 
It  should  be  borne  in  mind  that  where  the  relative  and  absolute 

1  See  paper  by  Eckart  to  which  reference  is  made  in  note  on  page  145, 
and  for  some  proportions  see  Marks'  Mech.  Eng.  Handbook,  page  1089. 


DESIGN  OF  THE  TANGENTIAL  WATER  WHEEL       209 

paths  coincide,  the  bucket,  as  at  A  (Fig.  124),  is  shown  in  its 
true  position  in  space.  But  a  bucket  as  at  B,  if  shown  only  in  its 
position  relative  to  the  jet,  has  its  true  position  in  space  to  the 
right  of  this.  This  may  be  seen  further  in  that  the  point  where 
the  relative  path  cuts  the  circle  to  the  right  corresponds  to  the 
actual  position  of  the  particle  of  water  as  it  leaves  the  right  hand 
side-  of  the  wheel  in  its  absolute  path.  The  use  of  these  curves 
will  enable  one  to  determine  the  best  shape  for  the  bucket  along 
the  lip  and  along  the  splitter,  as  well  as  the  best  outline  for  the 
notch. 

But  of  equal  importance  with  the  design  of  the  face  of  the 
bucket  is  that  of  the  shape  of  the  back.  As  the  bucket  A  enters 
the  jet  in  Fig.  124,  its  back  should  not  intersect  the  curve  of  the 
relative  path  of  the  water.  If  it  does  intersect  it,  it  indicates 
that  the  back  of  the  bucket  will  strike  the  water  in  the  jet  and 
it  is  obvious  that  this  would  result  in  a  loss  of  efficiency.  The 
back  of  the  bucket  could  strike  the  water,  despite  the  higher 
velocity  of  the  latter,  because  they  are  not  moving  in  the  same 
direction.  Hence  the  back  of  the  bucket  should  be  no  more 
than  tangent  to  the  curve  shown.  It  is  obvious  that  this  matter 
should  be  investigated  for  other  stream  lines  and  other  planes, 
besides  the  one  shown. 

Ideally  the  water  should  be  reversed  by  the  bucket  and  dis- 
charged backwards,  relatively,  at  an  angle  of  180°.  But  this  is 
impractical  because  the  water  would  then  be  unable  to  get  out 
of  the  way  of  the  next  bucket.  Hence  such  an  angle  should  be 
used  as  will  enable  the  water  to  be  discharged  with  an  absolute 
velocity  whose  lateral  component  is  sufficient.  As  has  been 
pointed  out,  the  closer  the  buckets  are  placed,  the  greater  must 
be  the  value  of  this  velocity  and  hence  the  more  this  angle  must 
be  made  to  differ  from  180°.  The  bucket  angle  used  in  practice 
is  about  170°. 

If  the  shape  of  the  bucket  can  be  determined  for  the  entrance 
and  discharge  edges  by  the  application  of  the  preceding  principles, 
the  bucket  may  be  completed  by  joining  these  two  portions  with 
any  smooth  surface  of  double  curvature.  There  should  be  no 
sharp  curvature  used  nor  anything  which  would  tend  to  cause 
any  abrupt  change  in  the  path  or  velocity  of  the  water. 

146.  Dimensions  of  Case. — The  case  should  be  made  of  suffi- 
cient size  to  allow  a  reasonable  clearance  between  it  and  the 
buckets  around  the  top  portion.  Usually  the  lower  portion  of 


210  HYDRAULIC  TURBINES 

the  wheel  is  below  the  floor  level  and  so  the  case  does  not  extend 
to  that  section.  But  in  any  event  there  should  be  ample  room 
on  either  side  of  the  buckets  here  so  that  the  water  discharged 
from  the  buckets  may  not  rebound  back  to  the  wheel.  It  is 
apparent  that  the  higher  the  head,  the  greater  the  discharge 
velocity,  and  hence  the  more  room  there  should  be  at  this  place, 
otherwise  water  will  be  thrown  back  upon  the  wheel  and  thus 
increase  the  so-called  windage  loss.  This  action  is  most  marked 
in  many  of  the  small  laboratory  wheels  that  have  been  made  with 
very  narrow  cases. 

147.  QUESTIONS  AND  PROBLEMS 

1.  How  may  the  diameter  of  a  Pelton  wheel  be  found  for  a  given  head, 
speed,  and  power?     How  may  the  diameter  of  jet  be  found? 

2.  What  are  the  principles  in  the  design  of  a  needle  nozzle?     How  may 
the  size  of  the  nozzle  tip  be  determined,  if  the  jet  diameter  is  given? 

3.  How  may  the  necessary  pitch  for  the  buckets  of  an  impulse  wheel 
be  computed? 

4.  Why  is  the  tangential  water  wheel  bucket  made  as  it  is  with  a  notch 
in  the  edge?    Would  it  be  possible  to  have  an  efficient  bucket  without  this? 

6.  How  may  the  specific  speed  of  a  Pelton  wheel  be  increased?    What 
limits  the  maximum  value  of  the  specific  speed? 

6.  How  may  the  shape  of  the  bucket  at  the  entrance  edge  be  determined? 
How  is  the  shape  of  the  entire  bucket  fixed? 

7.  Suppose  an  impulse  wheel  is  required  to  deliver  5,000  h.p.  at  300 
r.p.m.  under  a  head  of  1200  ft.     Find  the  diameter  of  jet  and  the  diameter 
of  wheel  necessary. 

8.  What  would  be  the  approximate  diameter  of  the  orifice  of  the  nozzle 
tip  in  problem  (7)  ? 

9.  What  would  be  the  probable  pitch  of  the  buckets  in  problem  (7)  and 
how  many  of  them  would  be  used  on  the  wheel? 

10.  The  bucket  for  the  wheel  in  problem  (7)  may  be  laid  out  on  the 
drafting  board. 


CHAPTER  XVI 
DESIGN  OF  THE  REACTION  TURBINE 

148.  Introductory.  —  Assume  that  the  head,  speed,  and  power 
for  a  proposed  turbine  are  known,  the  speed  and  power  of  the 
runner  having  been  so  chosen  as  to  give  the  specific  speed  neces- 
sary for  the  type  of  turbine  desired.     The  type  of  runner  will 
have  been  selected  in  accordance  with  the  principles  and  considera- 
tions of  the  preceding  chapters,  so  that,  as  the  problem  comes  to 
the  designer,  it  is  merely  a  matter  of  designing  a  turbine  to  fit 
the  specified  conditions. 

It  has  been  seen  that  practically  all  dimensions,  factors,  and 
even  characteristics  can  be  expressed  as  functions  of  the  specific 
speed,  hence  the  latter  is  the  logical  key  to  design.  After  the 
specific  speed  of  the  desired  unit  is  known,  the  proper  factors  may 
be  selected  in  the  light  of  previous  experience,  and  the  necessary 
dimensions  computed. 

The  data  given  in  this  chapter  is  to  be  understood  as  merely 
typical  of  present  practice.  It  is  perfectly  possible  to  alter  any 
of  the  quantities  given,  within  certain  limits,  providing  other 
related  factors  are  changed  also.  Consequently  runners  of  the 
same  specific  speed  may  be  built  without  their  being  identical  in 
all  other  respects.  Also,  of  the  numerous  variables,  certain  ones 
are  assumed  and  the  rest  computed  to  correspond.  It  is  appar- 
ent that  the  practice  of  designers  may  vary  according  to  what  is 
assumed  and  what  is  computed,  and  hence  the  procedure  given 
here  is  not  the  only  one  that  may  be  followed. 

149.  General  Dimensions.  —  As  explained  in  Art.  37,  the  value 
of  <t>e  increases  in  rational  design  as  the  specific  speed  increases. 
Customary  values  of  this  factor  for  different  values  of  N8  are 
given  by  a  curve  in  Fig.  126.     As  stated  in  the  preceding  article, 
this  curve  is  not  intended  to  be  followed  precisely,  but  the  varia- 
tion from  it  should  not  be  too  great. 

Having  selected  a  suitable  value  of  fa,  for  the  type  of  runner 
desired,  the  diameter  may  be  computed  from  equation  (52),  which 
reduces  to 


211 


212 


HYDRAULIC  TURBINES 


As  has  also  been  explained,  the  ratio  B/D  is  a  function  of  specific 
speed,  and,  choosing  a  value  of  this  from  Fig.  126,  the  height  of 
the  runner  at  entrance  or  the  height  of  the  guide  vanes  may 
at  once  be  determined. 

From  equations  (53)  and  (55)  we  may  find  the  value  of  cr, 
using  whichever  form  happens  to  be  more  convenient. 

6.01g 


Cr 


Cr   = 


BD\/h 
0.0000157N,2 


(65) 


(66) 


As  an  illustration  of  the  possible  variation  in  procedure,  it  may  be 
noted  that  we  compute  cr  after  assuming  the  value  of  B.     It 


Low  Ns 


C  - 


FIG.  125. 


would  be  equally  proper  to  assume  a  value  of  cr  and  compute  the 
corresponding  value  of  B.  It  should  also  be  noted  that  the  above 
factors  involve  the  assumption  that  5  per  cent,  of  the  total  area 
is  taken  up  by  the  runner  vanes.  After  the  design  has  progressed 
to  the  point  where  the  number  and  thickness  of  the  vanes  can  be 
determined,  the  above  may  be  corrected  if  that  refinement  is 
deemed  necessary. 

From  equation  (39),  letting  cu  =  ce  cos  on,  we  have 

cu  =  eh/2<f>e.  (67) 

In  the  case  of  high  specific  speed  runners  this  value  needs  to  be 
increased  about  5  per  cent.,  but  it  is  substantially  correct  for 
lower  speed  runners.  See  Art.  92. 


DESIGN  OF  THE  REACTION  TURBINE 


213 


100 


.150 


Ns  -  Metric 

200  250  300  350 


40  50  60  70 

Ns  -  English  Units 

FIG.  126. 


80 


100 


0.10 


130 
120 

no 

100 


I 


<a*  W 

•o 

a 

CO 

«    5° 
40 

30 
20- 
10 


20 


40  50  60 

Ns  -  English  Units 

FIG.  127. 


Extreme-High-»- 
400 


100 


214  HYDRAULIC  TURBINES 

The  direction  of  the  absolute  velocity  of  the  water  entering  the 
runner  may  now  be  found  since 

tan  ai  =  cr/cu  (68) 

and  the  direction  of  the  relative  velocity,  which  should  also  be 
the  direction  of  the  runner  vane,  is  given  by 

tan  0i  =  cr/(cu  -  <f>e).  (69) 

The  number  of  guide  and  runner  vanes  to  be  used  is  decided 
somewhat  arbitrarily,  but  one  fundamental  principle  to  be 
observed  is  that  they  should  not  be  equal  to  each  other  nor  any 
simple  multiple,  otherwise  pulsations  will  be  set  up.  For  sim- 
plicity of  design  and  shop  reasons  it  is  convenient  to  make  the 
guide  vanes  a  multiple  of  4.  Zowski's  rule  is  that  the  number 
of  guide  vanes,  n',  may  be  found  by 

n'  =  K'^/D  (70) 

where  K'  =  2.5  for  on  =  10°  to  20°,  3.0  for  on  =  20°  to  30°, 
3.5  for  on  =  30°  to  40°.  Although  K'  increases  with  the  specific 
speed,  the  diameter  of  the  runner  decreases  for  the  same  power  so 
that  actually  the  number  of  vanes  is  often  less. 

In  order  to  avoid  any  pulsations  the  runner  vanes  are  often 
made  an  odd  number,  though  other  designers  prefer  to  use  an 
eveji  number  which  is  2  less  than  the  number  of  guide  vanes. 
Zowski's  rule  is  that  the  number  of  runner  vanes,  n,  may  be 
fouid  by 

>^  n  =  K\/D  (71) 

-         j  *"""""! ""  i 

where K  =  3.7  for  a  low  specific  speed,  3.0  for  a  medium  specific 
speed,  and  2.2  for  a  high  specific  speed. 

160.  Profile  of  Runner. — The  profile  of  a  runner  is  shown  in 
Fig.,  125  and  the  notation  applied  to  it  is  clearly  indicated.  By 
Dz  is  meant  the  diameter  of  the  circle  passing  through  the  center 
of  gravity  of  the  outflow  area.  In  Fig.  34  were  shown  a  few 
typical  profiles,  and  a  more  complete  set  is  shown  in  Fig.  128. 

Tne_  exact  shape  of  profile  desired  is  determined  largely  by 
experience,  a  shape  being  used  that  had  been  found  satisfactory 
for  the: specific  speed  in  question.  But  it  is  also  a  matter  of  the 
whim  or  taste  of  the  designer,  as  theory  has  little  bearing  on  it 
directly.  But  the  theory  (Art.  66)  does  indicate  that  a  very 
sharp  radius  of  curvature  is  undesirable  and  an  excessive  curva- 
ture near  the  band,  as  is  often  found  with  certain  high  specific 


DESIGN  OF  THE  REACTION  TURBINE 


215 


speed  runners,  simply  results  in  the  water  failing  to  follow  the 
path  desired,  which  may  result  in  eddy  losses,  thus  not  only  re- 
ducing the  efficiency  but  also  facilitating  corrosion  at  such  places. 
It  is  also  considered  desirable  to  have  the  length  of  the  path 
along  the  crown  approximately  equal  to  that  along  the  band, 
though  it  is  frequently  a  little  more. 

With  the  values  given  by  the  curves  in  Fig.  126  and  the  aid 
of  the  samples  shown  in  Fig.  128  and  elsewhere,  it  is  possible  to 
lay  out  a  profile  that  should  be  satisfactory,  But  before  draw- 
ing in  the  outflow  edge,  it  is  necessary  to  consider  the  stream 
lines.  Let  us  assume  that  all  particles  of  water  flow  with  the 


Ns=10 


Ns=30 


Ns  =  70  Ns=85 

FIG.  128. — Typical  profiles. 


Ns  =  100 


same  velocities  and  that  the  total  rate  of  discharge  is  divided  into 
equal  portions.  If  then  the  water  passages  are  divided  into 
portions  of  equal  cross-section  area,  it  follows  that  the  boundary 
lines  between  them  must  be  stream  lines.  Hence  the  height  B 
at  entrance  may  be  divided  into  equal  portions,  and  for  our  pur- 
pose here  we  shall  assume  four.  Also  the  draft  tube  may  be 
divided  into  concentric  rings  of  equal  area,  as  shown  in  Fig.  125, 
the  section  CC  being  removed  far  enough  from  the  runner  for 
the  stream  lines  to  have  become  parallel.  Then  the  curves  in 
between  may  be  sketched  in  by  eye.  It  may  be  noted  that,  if  at 
an  intermediate  section  a  line  be  drawn  normal  to  these  stream 
lines,  the  product  r  A  b  must  be  constant  along  this  normal. 


216  HYDRAULIC  TURBINES 

In  reality  the  velocity  is  not  the  same  for  all  stream  lines, 
those  nearer  the  band  having  a  higher  velocity  than  those  near 
the  crown,  because  of  the  smaller  radius  of  curvature.  The 
difference  is  very  small  in  the  case  of  the  low  speed  runner,  but 
for  the  high  speed  runner  the  velocity  near  the  crown  may  be 
about  30  per  cent,  below  the  mean  and  that  near  the  band  about 
50  per  cent,  above  the  mean  velocity.  This  lowers  the  flow  lines 
at  entrance  below  the  positions  as  determined  above.  Also  the 
water  in  the  draft  tube,  as  in  any  other  pipe,  tends  to  flow  with 
a  higher  velocity  in  the  center  than  around  the  circumference. 
In  accordance  with  these  considerations  the  tentative  flow  lines, 
as  first  sketched  in,  may  be  modified,  according  to  judgement. 
If  further  refinement  is  desired,  this  second  set  of  flow  lines  may 
be  checked  and  corrected  by  the  method  given  in  Appendix  B. 

The  outflow  edge  may  now  be  drawn  by  making  it  perpendicu- 
lar to  these  various  stream  lines.  But  for  the  portion  near  the 
crown  this  procedure  may  tend  to  bring  the  discharge  edge  too 
close  to  the  axis  of  rotation.  The  theory  (Art.  91)  indicates 
that  a  large  variation  in  the  radii  to  points  along  the  outflow 
edge  is  undesirable,  and  that  the  discharge  edge  should  really  be 
parallel  to  the  axis  of  rotation.  The  latter  is  not  practicable, 
but  for  this  portion  of  the  outflow  edge  a  compromise  is  effected 
by  making  it  about  a  mean  between  a  line  parallel  to  the  axis 
and  one  which  would  be  normal  to  all  the  stream  lines. 

It  should  be  noted  that  in  these  profiles  we  are  dealing  with 
circular  projection,  by  which  is  meant  that  points  are  rotated 
about  the  axis  of  the  runner  until  they  lie  in  the  plane  of  the  paper. 
Thus  the  actual  stream  lines  are  not  as  shown  in  such  a  view, 
the  lines  drawn  being  merely  circular  projections  of  the  actual 
paths. 

151.  Outflow  Conditions  and  Clear  Opening. — In  case  a  stream 
line  is  not  perpendicular  to  the  outflow  edge  it  indicates  that  the 
relative  velocity  of  the  water  leaving  the  runner  is  not  really 
normal  to  the  outflow  area,  as  the  latter  would  ordinarily  be 
measured.  It  is  thus  more  convenient  to  deal  with  components 
of  the  velocity  in  a  plane  normal  to  the  discharge  edge  at  the  point 
in  question.  Referring  to  Fig.  129,  let  us  assume  that  the  out- 
flow edge  is  actually  in  the  plane  of  the  paper.  If  <*2  =  90°  be 
assumed  to  be  the  conditions  for  which  the  runner  is  to  be  de- 
signed, the  absolute  velocities  of  the  water  at  all  points  along 
the  outflow  edge  also  lie  in  the  plane  of  the  paper  but  have  the 


DESIGN  OF  THE  REACTION  TURBINE          217 

various  directions  indicated  by  the  stream  lines.  The  magni- 
tude of  the  absolute  velocity  in  the  draft  tube  may  be  computed 
from  the  rate  of  discharge  and  the  area  of  the  tube,  provided  there 
is  no  whirl.  The  absolute  velocity  of  the  water  discharging  from 
the  runner  vanes  is  somewhat  more  than  this  because  of  the  space 
taken  up  by  the  vanes,  hence  the  Vz  at  discharge  from  the  runner 
should  be  larger  than  that  in  the  tube  by  a  factor  which  may  be 
assumed  to  be  about  15  per  cent.  The  absolute  velocity  of  the 
water  is  ordinarily  assumed  to  be  uniform  all  along  the  outflow 
edge,  but  actually  there  may  be  some  variation  in  it  in  certain 
types  of  runner  because  of  the  varying  radii  of  curvature  of  the 
different  stream  lines. 

Values  of  V%  may  be  laid  off  along  their  respective  stream  lines 
as  indicated  in  Fig.  129,  and,  where  they  are  not  perpendicular 
to  the  stream  line,  components  indicated  by  TV  should  be  found. 
The  latter  will  be  used  in  the  diagram  below. 

The  linear  velocity  of  a  point  on  the  runner  may  be  laid  off  at 
any  radius  perpendicular  to  a  line  representing  this  radius  and 
a  third  line  then  drawn  so  as  to  form  a  triangle.  It  is  often 
convenient  to  lay  off  Ui  at  radius  r1}  as  shown  in  Fig.  129.  At 
any  other  radius  the  peripheral  velocity  is  given  by  the  inter- 
cept. Since  a2  =  90°  and  F2  (or  F2')  is  known  for  each  stream 
line,  a  velocity  diagram  may  be  drawn  with  u2  as  a  base,  and  as 
many  of  these  constructed  as  there  are  stream  lines.  It  should 
be  noted  that  the  crown  and  band  of  the  runner  form  boundaries 
and  hence  furnish  stream  lines  also.  The  series  of  diagrams  so 
constructed,  as  in  Fig.  129,  give  the  values  of  the  relative 
velocities  and  the  direction  of  the  runner  vane  at  outflow  for 
various  points  along  its  edge,  if  the  absolute  velocity  of  dis- 
charge is  to  be  at  90°. 

The  clear  opening  is  the  shortest  distance  from  a  point  on  the 
discharge  edge  of  one  vane  to  the  back  of  the  next  vane.  This 
is  shown  in  Fig.  129  and  it  may  be  seen  that  the  clear  opening  is 
practically  equal  to  pitch  X  sin  £'2  —  vane  thickness.  The  pitch 
at  any  radius  is  known,  since  the  number  of  runner  vanes  are 
known.  It  may  conveniently  be  found  by  laying  off  a  value  for 
the  pitch  at  some  radius,  similar  to  the  procedure  for  the  veloci- 
ties above,  and  then  the  pitch  at  any  other  radius  may  be  found 
by  using  the  proper  intercept.  This  diagram  should  be  drawn 
to  the  same  scale  as  the  runner.  The  angle  /3'2  is  that  found  in 
the  velocity  diagrams  constructed  by  using  values  of  V2.  In 


218 


HYDRAULIC  TURBINES 


case  this  is  not  necessary,  the  angle  of  course  becomes  merely  /32. 
Referring  to  Fig.  129  again,  it  may  be  seen  that  the  clear  opening 
may  be  found  graphically  at  any  point  by  laying  off  a  line  from 
the  end  of  the  pitch  distance  perpendicular  to  the  vector  t/2 


Fia.  129. — Determination  of  clear  opening  at  outflow. 

(or  v2).     The  vane  thickness  may  be  subtracted  from  this  line 
and  the  remainder  is  the  clear  opening. 

The  clear  opening,  determined  for  each  one  of  the  stream  lines 
by  the  method  just  described,  may  be  laid  off  from  the  outflow 
edge  of  the  runner  in  the  form  of  an  arc.  The  curve  enveloping 


DESIGN  OF  THE  REACTION  TURBINE          219 

these  arcs  represents  the  clear  opening.  The  clear  opening  is  not 
actually  in  the  plane  of  the  paper  but  each  element  of  it  is  as- 
sumed to  be  rotated  about  the  line  representing  the  outflow  edge 
until  it  does  lie  in  the  plane  of  the  paper.  This  may  introduce 
some  error  where  the  curvature  of  the  outflow  edge  is  very 
marked.  Since  the  angle  through  which  it  is  rotated  is  usually 
not  very  great,  the  error  is  small  and  hence  this  area  in  the 
plane  of  the  paper  may  be  taken  as  the  true  outflow  area  between 
two  runner  vanes. 

The  rate  of  discharge  through  any  section  of  the  outflow  area 
may  now  be  determined  by  multiplying  each  sectional  area  by 
the  average  value  of  the  component  of  the  relative  velocity 
through  it.  In  reality  the  true  outflow  area  is  normal  to  the 
true  relative  velocity,  but  in  case  the  latter  is  not  in  the  plane 
perpendicular  to  outflow  edge,  we  obtain  the  same  product  by 
the  method  used. 

The  rate  of  discharge  for  the  turbine  may  now  be  computed  as 

q  =  c  nSt/2Aa2  (72) 

where  n  denotes  the  number  of  vanes,  v'z  the  average  value  of  the 
relative  velocity  for  the  section  considered  and  in  the  plane 
defined,  and  Aa2  the  element  of  area  between  two  stream  lines. 
The  coefficient  of  discharge  may  be  taken  as  0.95  for  low  specific 
speed  runners,  0.90  for  medium  specific  speed  runners,  and  0.85 
for  high  specific  speed  runners.  This  is  really  a  coefficient  of 
contraction,  since  the  actual  stream  areas  are  less  than  the  areas 
at  the  end  of  the  converging  passages. 

If  the  rate  of  discharge  is  not  the  exact  quantity  required,  the 
outflow  area  may  be  altered  somewhat  by  shifting  the  position 
of  the  crown  or  by  changing  the  outflow  edge  until  the  desired 
result  is  obtained.  It  may  be  noted  that  the  friction  in  flow 
through  the  runner  passages  will  be  less  along  the  middle  stream 
lines  than  for  those  near  the  crown  and  band.  The  relative 
velocity  will  thus  be  a  little  higher  in  the  middle  and  to  preserve 
a  " radial"  discharge  all  along  the  edge  it  will  be  necessary  to 
decrease  the  angle  02,  which  can  be  done  by  increasing  the  clear 
opening  a  trifle  along  the  middle  portion.  By  the  same  line  of 
reasoning  the  opening  may  be  reduced  slightly  at  each  end. 

152.  Layout  of  Vane  on  Developed  Cones. — By  the  methods 
previously  given,  the  conditions  at  entrance  to  and  discharge 
from  a  turbine  runner  may  be  determined.  Theory  does  not 


220 


HYDRAULIC  TURBINES 


prescribe  the  form  of  vane  between  these  two  portions,  except 
that  it  must  be  a  smooth  surface  of  gradual  curvature  to  avoid 
eddy  losses.  In  order  to  determine  the  form  of  this  surface,  it 
is  convenient  to  lay  out  the  vanes  on  developed  cones. 

In  Fig.  130  are  shown  several  lines  which  are  elements  of  cones 
whose  axes  coincide  with  the  axis  of  rotation  of  the  runner. 


These  elements  should  be  so  taken  as  to  cut  both  the  discharge 
and  the  entrance  edges  and  it  is  desirable  also  to  have  them 
approximately  perpendicular  to  the  outflow  edge.  If  this  re- 
quirement in  some  instances  causes  the  vertex  of  a  cone  to  be 
removed  to  too  great  a  distance,  a  cylinder  may  be  used  instead, 
the  cylinder  being  a  special  case  of  the  cone. 

It  is  desirable  to  begin  with  the  cone  nearest  the  crown,  as 


DESIGN  OF  THE  REACTION  TURBINE 


221 


cone  A  in  Fig.  130.  This  is  developed  in  Fig.  131,  the  distances 
OM  and  ON  being  equal  respectively  in  both  figures.  Along  the 
arc  through  M  is  laid  off  a  pitch  distance  MP,  corresponding  to 
the  actual  pitch  of  the  runner  at  M.  (This  is  the  pitch  at  the 
radius  of  M  from  the  axis  of  rotation  and  not  at  the  radius  OM) . 


FIG.   131. — Layout  of  vanes  on  developed  cones. 

From  P  are  described  two  arcs  whose  radii  are  equal  to  the  clear 
opening  at  N  and  the  clear  opening  plus  the  vane  thickness. 
From  M  the  two  sides  of  a  vane  are  then  drawn  tangent  to  these 
arcs,  as  shown.  The  actual  vane  is  sharpened  on  the  end,  so  as 
to  minimize  eddy  losses.  The  front  side  of  the  vane  from  M 
tangent  to  the  outer  arc  should  be  practically  a  straight  line.  In 


222 


HYDRAULIC  TURBINES 


theory  the  portion  along  the  back  of  the  vane  as  far  as  point  S 
should  be  a  spiral  of  the  form  r  =  constant  +  KB'  (K  being  a 
constant  and  0'  the  angle  subtended  at  0),  or  an  involute  so  as 
to  keep  the  cross-section  area  of  the  stream  constant  from  PS 
to  the  arc  MP.  Actually  this  is  not  usually  convenient,  but  the 
actual  shape  is  between  such  a  curve  and  a  straight  line.  Thus 
the  shape  of  discharge  ends  of  the  vanes  are  completely  fixed. 
It  may  be  noted  that  the  clear  opening  PS  is  a  constant  distance 
for  any  cone  taken  through  point  M  in  Fig.  130. 

In  the  case  of  the  entrance  ends  of  the  vanes  we  might  proceed 
in  the  same  way,  but  it  is  usually  more  convenient  to  lay  off  the 
vane  angle  instead.  It  may  be  noted  however  that  the  value  of 


FIG.  132. 

the  angle  on  the  developed  cone  is  different  for  different  cones 
through  the  same  point.  In  the  most  general  case  the  entrance 
edge  of  a  runner  may  be  inclined  to  the  axis  of  rotation  at  an 
angle  7  as  shown  in  Fig.  132,  and  also  it  may  not  be  in  the  same 
plane  as  the  axis  but  in  another  plane  at  an  angle  e.  Let  the 
elements  of  the  cone  make  an  angle  6  with  the  axis,  while  the 
projected  stream  line  at  entrance  makes  an  angle  6  with  the 
axis.  It  may  be  noted  that  the  actual  velocity  diagram  should 
always  be  in  the  plane  of  the  stream  line,  and  is  not  necessarily 
in  a  plane  perpendicular  to  the  axis.  If  the  angle  of  the  relative 
velocity  ft  becomes  0"i  on  the  developed  coWe,  it  may  be  proven 
by  geometry  and  trigonometry  that 

sin  (8  +  T) 

sin  (B  +  7) 


tan  /3"i  =  tan  ft 


sin  (0  - 


(73) 


DESIGN  OF  THE  REACTION  TURBINE          223 

If  the  angle  e  is  zero,  as  it  is  in  many  turbine  runners,  the  expres- 
sion in  the  brackets  becomes  equal  to  unity  and  the  above 
formula  is  greatly  simplified. 

The  angle  (3"i  may  be  laid  off  as'at  Rf  in^Fig.  131*and  the  vane 
tip  then  moved  along  until  it  reaches  such  a  position  that  a  good 
smooth  curve  may  join  this  end  of  the  vane  with  the  portion  MS. 
The  complete  vane  is  now  MR.  The  next  vane  PQ  should  be 
drawn  in  order  to  find  out  if  the  cross-section  area  continuously 
decreases  from  entrance  to  outflow. 

A  similar  procedure  may  be  gone  through  with  for  the  other 
cones  or  cylinders,  except  for  one  restriction.  The  relative 
position  of  M  and-R  is  purely  arbitrary  in  the  first  cone  used, 
but  for  all  the  remainder  this  much  is  fixed.  Psually  runners 
are  so  constructed  that  all  points  along  the  outflow  edge  are  in 
one  plane  and  furthermore  this  plane  contains  the  axis  of  rota- 
tion. Our  discussion  will  therefore  be  confined  to  this  case, 
though  the  method  could  readily  be  extended  to  the  more  general 
treatment,  if  desired.  If  all  points  along  the  entrance  edge  are 
in  the  same  plane  and  this  plane  contains  the  axis  (the  angle  e 
being  zero),  the  arc  NR  subtends  an  equal  number  of  pitches  or 
fractions  thereof  in  every  cone  or  cylinder.  This  is  most  con- 
veniently laid  off  on  the  drawing  board  by  establishing  an  arc 
at  a  fixed  radius  from -the  axis  of  rotation  and  putting  this  in 
every  cone.  Its  length  is  the  same  in  every -case.  It  is  conven- 
ient to  take  this  arc  through  the  point  where  the  diameter  D  is 
measured. 

If  the  entrance  edge  is  inclined,  as  indicated  by  the  angle  e 
in  Fig.  132,  the  runner  vane  near  the  crown  subtends  a  greater 
angle  than  that  portion  nearer  the  band,  and  a  different  length 
of  arc  is  used  in  different  cones. 

If  it  is  difficult  or  impossible  to  secure  proper  vane  curves 
in  some  of  the  cones,  it  may  be  necessary  to  go  back  to  cone  A  and 
to  change  the  position  of  R  so  that  the  vane  MR  subtends  a 
different  angle.  One  advantage  of  inclining  the  entrance  edge, 
so  that  it  makes  the  angle  as  shown,  is  that  it  permits  of  securing 
better  vane  curves  in  all  the  cones,  in  some  instances.  This  is 
particularly  true  in  the  case  of  high  specific  speed  runners. 

153.  Intermediate  Profiles. — The  various  cones  and  cylinders 
of  Fig.  131  are  next  divided  up  into  fractional  pitches,  preferably 
quarter  pitches.  These  lines  represent  a  series  of  planes  passing 
through  the  axis  and  cutting  the  vane.  The  model,  'the  photo- 


224 


HYDRAULIC  TURBINES 


graph  of  which  is  shown  in  Fig.  133,  illustrates  this  very  nicely 
and  shows  the  intermediate  profiles  cut  out.  For  cone  A  the 
distances  from  0  to  the  intersections  of  planes  I,  II,  III,  IV,  etc. 
with  the  vane  are  transferred  from  Fig.  131  to  Fig.  130,  being  laid 
off  from  0  along  the  line  OMN.  A  similar  procedure  is  followed 
for  all  the  cones  and  cylinders. 

Since  planes  I,  II,  III,  IV,  etc.  are  the  same  in  every  cone, 
it  follows  that  profiles  can  be  drawn  through  all  the  points  with 
the  same  number  in  Fig.  130.  If  the  vane  surface  is  proper, 


FIG.  133. — Model  constructed  by  Lewis  F.  Moody  illustrating  development  of 

vane  surface. 


these  profiles  will  all  be  smooth  curves,  and  will  all  be  similar 
but  changing  gradually  from  the'jentrance^edge  to  the  discharge 
edge.  If  the  curves  are  not  smooth  and  of  the  proper  shape,  it 
will  be  necessary  to  change  the  vanes  laid  out  on  the  developed 
cones  until  both  the  profiles  and  the  curves^on  all  thejcones  are 
satisfactory.  Thus  these  profiles  serve  as^a  check  on  the  work, 
and  also  are  desirable  in  order  to  determine  the  pattern  maker's 
sections. 


DESIGN  OF  THE  REACTION  TURBINE 


225 


154.  Pattern  Maker's  Sections. — If  a  plane  be  passed  through 
Fig.  130  normal  to  the  axis  of  the  runner,  the  vane  will  cut  a 
curve  in  it  which  may  be  found  as  follows.  In  Fig.  134  a  portion 
of  this  plane  is  drawn  and  it  is  subdivided  into  the  same  fractional 
pitches  as  the  various  developed  cones.  The  distance  XY  may 
be  transferred  from  Fig.  130  to  Fig.  134  and  laid  off  in  plane  7. 


FIG.  134. — Patternmaker's  sections. 

The  distance  XZ  may  be  transferred  in  similar  manner  and  laid 
off  along  plane  //.  Proceeding  in  this  way  the  entire  curve  may 
be  established. 

A  series  of  parallel  planes  at  fixed  distances  apart  are  used 
and  curves  constructed  for  all  of  them  in  the  same  manner,  but 
on  the  same  drawing,  as  in  Fig.  134.  Where  the  curvature  of 
the  vane  is  sharper  the  planes  are  spaced  closer  together,  as  shown 

15 


226 


HYDRAULIC  TURBINES 


in  Fig.  133.  These  planes  really  represent  the  surfaces  of  boards 
of  different  thicknesses  and  on  each  board  the  proper  curve  from 
Fig.  134  may  be  laid  out.  These  boards,  when  placed  together, 
as  in  Fig.  135,  and  the  surface  smoothed  down,  give  the  proper 
shape  of  the  vane  surface.  In  this  way  the  core  box  for  the  vane 
may  be  formed. 

Since  the  vane  has  both  a  front  and  a  back  surface  which  differ 
slightly  from  each  other,  this  entire  proceeding  is  carried  through 


FIG.  135.- 


(Courtesy  of  Wellman-Seaver-M organ  Co.) 
-Construction  of  pattern  for  rear  face  of  core  box  for  runner  vane. 


for  both  surfaces.  It  is  desirable  to  draw  the  profiles  and  pattern 
maker's  sections  for  the  front  of  the  vane  in  full  lines  and  for  the 
back  of  the  vane  as  dotted  lines.  Also  if  the  lines  for  the  back 
of  the  vane  in  Fig.  134  are  placed  one  pitch  distance  away  from 
those  for  the  front,  one  can  more  readily  grasp  the  appearance  of 
the  passageway  between  the  two  vanes. 


DESIGN  OF  THE  REACTION  TURBINE 


227 


155.  The  Case  and  Speed  Ring. — The  velocity  of  the  water  in 
a  spiral  case  may  range  from  0.15  \/Zgh  to  0.20  -\/2gh,  the  higher 
factor  being  used  for  lower  heads.  For  globe  and  cylinder  cases 
even  lower  velocities  should  be  employed,  because  of  the  hydraulic 
losses  in  such  cases.  A  spiral  case  should  be  so  proportioned  that 
equal  quantities  of  water  flow  to  equal  portions  of  the  runner,  as 
shown  in  Fig.  136.  If  the  cross-section  of  the  case  is  circular, 
it  may  be  seen  that  the  radius  of  a  point  on  the  outer  boundary 
is  given  by  r  =  \/cd  +  K  where  c  and  K  are  constants  and  6 
the  subtended  angle,  as  this  curve  will  give  an  area  which  is 
directly  proportional  to  the  angle.  For  any  other  change  of 
cross  section,  it  is  easy  to  determine  the  form  necessary  by  apply- 
ing the  principle  that  the  area  must  vary  as  the  subtended  angle 
at  the  runner  axis. 


FIG.  136. 

The'path  of  a  free  stream  line  in  the  case  may  be  plotted  by 
the  principles  of  Art.  66.  If  Vc  be  the  velocity  with  which  the 
water  enters  the  case  and  at  radius  rc,  the  tangential  component 
of  the  velocity  at  any  other  radius  is  given  by 

Vu  =  rc  Vc/r 

while  the  radial  component  is  given  by 

Vr  =  q  /2irrb 

where  r  is  the  radius  in  feet  to  any  point  and  b  the  height  of  the 
water  passage  in  feet.  These  two  components  give  the  direction 
of  the  water  at  any  radius,  and  by  sketching  in  a  series  of  tangents 
it  is  easy  to  plot  the  path  by  a  little  trial. 

If  speed  ring  vanes  are  used,  they  should  be  so  shaped  as  to 


228 


HYDRAULIC  TURBINES 


conform  to  these  free  stream  lines.  The  number  of  speed  ring 
vanes  should  be  half  that  of  the  number  of  guide  vanes. 

156.  The  Guide  Vanes. — The  guide  vanes  should  be  so  shaped 
that  they  are  tangent  to  the  free  stream  lines  of  the  water  entering. 
If  the  turbine  is  set  in  an  open  flume  or  case  where  free  stream  lines 
cannot  readily  be  plotted,  the  guide  vanes  are  made  so  as  to 
approach  a  radial  direction  at  this  point. 

The  direction  of  the  water  is  changed  during  flow  through  the 
guide  passages  from  that  of  the  free  stream  line  to  the  direction 
desired.  After  a  particle  of  water  passes  point  a  in  Fig.  137,  its 
direction  should  remain  unchanged  until  it  strikes  the  runner,  as 
it  is  now  following  a  free  stream  line  once  more.  Since  the  space 
between  guide  vanes  and  runner  vanes  is  one  of  a  uniform  height, 
it  may  be  shown  that  free  stream  lines  are  then  equi-angular  or 


FIG.  137. 

logarithmic  spirals.  Thus  the  portion  of  the  vane  from  a  to  & 
should  be  such  a  curve.  The  other  side  of  the  vane  may  be 
either  a  straight  line  or  another  logarithmic  spiral.  The  equation 
of  the  equi-angular  spiral  is  loge  r  =  0  tan  a,  where  a  is  the  angle 
desired  and  6  is  the  subtended  angle. 

There  should  be  considerable  clearance  between  the  ends  of 
the  guide  vanes  and  the  runner  vanes  so  that  the  streams  of  water 
from  the  guides  may  unite  into  a  solid  ring  before  entering  the 
runner.  In  particular  the  point  of  intersection  o  of  Fig.  137 
should  be  located  outside  the  runner,  so  that  no  eddies  maybe 
produced  unnecessarily  in  the  latter. 

The  gates  are  sometimes  pivoted  near  the  point  so  that,  if  the 
governor  mechanism  fails,  the  gates  will  drift  shut.  But  this 
places  the  vane  shaft  at  a  point  where  the  section  of  the  vane 


DESIGN  OF  THE  REACTION  TURBINE          229 

should  be  small,  and  also  makes  it  necessary  to  exert  a  consider- 
able torque  to  hold  them  wide  open.  The  better  practice  is  to 
so  locate  the  pivot  that  x/y  =  3/2.  The  gates  are  then  hydrauli- 
cally  balanced  when  about  half  way  open  and  the  torque  resuired 
for  either  extreme  position  is  less. 

157.  QUESTIONS  AND  PROBLEMS 

1.  Given  the  head,  speed  and  power  for  a  reaction  turbine,  how  may  the 
size  of  the  runner,  the  height  of  the  guide  vanes,  and  the  diameter  of  the 
draft  tube  be  determined? 

2.  For  the  case  in  problem  (1),  how  would  the  guide  vane  angle  and  the 
runner  vane  angle  be  determined?     What  principles  are  involved  in  deciding 
upon  the  number  of  guide  vanes  and  runner  vanes? 

3.  How  is  the  profile  of  a  runner  to  be  fully  determined?     How  should  the 
stream  lines  be  drawn  in? 

4.  How  may  the  clear  opening  of  a  turbine  runner  be  determined? 

5.  How  may  the  capacity  of  a  runner  be  checked?     What  changes  can 
be  made  in  order  that  its  capacity  may  be  exactly  that  desired? 

6.  How  are  runner  vanes  laid  out  on  developed  cones? 

7.  Having  the  vanes  laid  out  on  developed  cones,  how  may  the  inter- 
mediate profiles  be  constructed?     What  use  is  made  of  these? 

8.  How  are  pattern  maker's  sections  drawn? 

9.  What  is  the  object  of  plotting  the  free  stream  lines  in  a  spiral  case? 

10.  How  should  guide  vanes  be  shaped?     What  other  factors  should  be 
considered  in  their -design? 

11.  A  turbine  runner  is  to  deliver  4000  h.p.  at  600  r.p.m.  under  a  head  of 
305  ft.     Determine  D,  B,  Dt,  D*,  «i,  /3'i,  V2,  number  of  guide  vanes,  and 
number  of  runner  vanes. 

Ans.     D  =  37  in.,  B  =  6.67  in.,  Dt  =  34.4  in.,  Dd  =  33.3  in.,  ttl  =  16°, 
p\  =  102°,  F2  =  25.6  ft.  per  second,  24  guide  vanes,  and  22  runner  vanes. 

12.  A  turbine  runner  is  to  be  designed  for  2000  h.p.  at  300  r.p.m.  under  a 
head  of  88  ft.     Find  same  as  in  problem  (11). 

Ans.     D  =  43.2  in.,  B  =  16.4  in.,  «i  =  20°,  ffl  =  125°,  20  guide  vanes, 
and  18  runner  vanes. 

13.  A  runner  is  to  be  designed  to  deliver  3000  h.p.  at  200  r.p.m.  under  a 
head  of  64  ft.     Find  the  results  called  for  in  problem  (11). 

14.  Find  the  allowable  height  above  the  tail  water  level  for  each  of  the 
runners  in  the  preceding  three  problems. 

15.  Draw  profile,  sketch  tentative  flow  lines,  construct  velocity  diagrams, 
lay  out  vanes  on  cones,  and  draw  patternmaker's  sections  for  one  of  the 
turbines  given  above. 


CHAPTER  XVII 
CENTRIFUGAL  PUMPS 

158.  Definition. — Centrifugal  pumps  are  so  called  because  of 
the  fact  that  centrifugal  force  or  the  variation  of  pressure  due  to 
rotation  is  an  important  factor  in  their  operation.  However, 
as  will  be  shown  later,  there  are  other  items  which  enter. 

The  centrifugal  pump  is  closely  allied  to  the  reaction  turbine 
and  may  be  said  to  be  a  reversed  turbine  in  many  respects. 
Therefore  it  will  be  found  that  most  of  the  general  principles 
given  in  Chapter  VII  will  apply  here  also  with  suitable  modifica- 
tions. Energy  is  now  given  up  by  the  vanes  of  the  impeller 


FIG.  138. — Turbine  pump. 


FIG.  139. — Volute  pump. 


to  the  water  and  we  have  to  deal  with  a  lift  instead  of  a  fall. 
The  direction  of  flow  through  the  impeller  is  radially  outward. 
During  this  flow  both  the  pressure  and  the  velocity  of  the  water 
are  increased  and  when  the  water  leaves  the  impeller  a  large 
part  of  its  energy  is  kinetic.  In  any  efficient  pump  it  is  neces- 
sary to  conserve  this  kinetic  energy  and  transform  it  into  pressure. 
159.  Classification. — Centrifugal  pumps  are  broadly  divided 
into  two  classes: 

1.  Turbine  Pumps. 

2.  Volute  Pumps. 

230 


CENTRIFUGAL  PUMPS 


231 


While  there  are  other  types  besides  these,  the  two  given  are 
the  most  important,  and  only  these  will  be  considered  in  this- 
chapter. 

The  turbine  pump  is  one  in  which  the  impeller  is  surrounded  by 
a  diffusion  ring  containing  diffusion  vanes.  These  provide 
gradually  enlarging  passages  whose  function  is  to  reduce  the 
velocity  of  the  water  leaving  the  impeller  and  efficiently  trans- 
form velocity  head  into  pressure  head.  The  casing  surrounding 
the  diffuser  may  be  either  circular  as  shown  in  Fig.  138  or  it  may 
be  of  a  spiral  form.  This  latter  arrangement  would  be  similar 
to  that  of  the  spiral  case  turbine  shown  in  Fig.  55. 

The  volute  pump  is  one  which  has  no  diffusion  vanes,  but, 
instead,  the  casing  is  of  a  spiral  type  so  made  as  to  gradually 
reduce  the  velocity  of  the  water  as 
it  flows  from  the  impeller  to  the  dis- 
charge pipe.  (See  Fig.  139.)  Thus 
the  energy  transformation  is  ac- 
complished in  a  different  way.  The 
spiral  curve  for  such  a  case  is  usu- 
ally called  the  volute,  and  from  this 
the  pump  receives  its  name. 

The  discussion  of  the  volute  pump 
will  apply  equally  well  to  all  other 
types  without  diffusion  vanes.  The 
only  difference  will  be  that  these 
other  types  are  less  efficient  and  also  it  will  probably  be  im- 
possible to  express  the  shock  loss  at  exit  in  any  satisfactory  way. 
Some  of  these  other  types  have  circular  cases  with  the  impeller 
placed  either  concentric  or  eccentric  within  them.  Their  only 
merit  is  cheapness. 

160.  Centrifugal  Action. — If  a  vessel  containing  water  or  any 
liquid  is  rotated  at  a  unifrom  rate  about  its  axis,  the  water  will 
tend  to  rotate  at  the  same  speed  and  the  surface  will  assume  a 
curve  as  shown  in  Fig.  140.  This  curve  can  be  shown  to  be  a 
parabola  such  that  h  =  Uz2/2g,  where  u^  =  linear  velocity  of 
vessel  at  radius  r2.  If  the  water  be  confined  so  that  its  surface 
cannot  change,  the  pressure  will  follow  the  same  law,  as  shown 
in  Art.  65. 

If,  as  in  Fig.  141,  we  have  the  water  in  a  closed  chamber  set  in 
motion  by  a  paddle  wheel,  the  pressure  in  an  outer  chamber 
communicating  with  it  will  be  greater  than  that  in  the  center 


FIG.   140. 


232 


HYDRAULIC  TURBINES 


by  the  amount  u22/2g.  If  a  piezometer  tube  be  inserted  in  this 
chamber,  water  will  rise  in  it  to  a  height  h  =  uz*/2g.  If  the 
height  of  the  tube  be  somewhat  less  than  this,  water  will  flow 
out,  and  we  would  have  a  crude  centrifugal  pump. 

161.  Notation. — The  notation  used  will  be  essentially  the  same 
as  that  for  the  turbine.  To  this,  however,  we  shall  add  a'2 
as  the  angle  the  diffusion  vanes  make  with  uz,  and  subscript  (3)  to 
denote  a  point  in  the  casing. 


FIG.  141. 


The  actual  lift  of  the  pump  will  be  denoted  by  h,  while  the  head 
that  is  imparted  to  the  water  by  the  impeller  will  be  denoted  by 
h".  If  h'  represents  all  the  hydraulic  losses  within  the  pump  and 
eh  represents  the  hydraulic  efficiency,  we  may  write 


ehh"  =  h"  -  h' 


(74) 


It  will  also  be  found  to  be  more  convenient  to  express  all  veloc- 
ities in  terms  of  u2  and  v%. 

Whereas  turbines  are  rated  according  to  the  diameter  of  the 
runner,  centrifugal  pumps  are  rated  according  to  the  diameter 
of  the  discharge  pipe  in  inches.  The  usual  velocity  of  flow  at  the 
discharge  is  10  ft.  per  second.  From  this  the  size  of  pump  nee- 


CENTRIFUGAL  PUMPS  233 

essary  for  a  given  capacity  may  be  approximately  estimated. 
In  some  cases,  however,  the  velocity  may  be  twice  this  value. 

162.  Definition  of  Head  and  Efficiency.  —  In  all  cases  the  head 
h  under  which  the  pump  operates  is  the  actual  vertical  height 
the  water  is  lifted  plus  all  losses  in  the  suction  and  discharge 
pipes.  It  should  be  noted  that  the  velocity  head  at  the  mouth 
of  the  discharge  pipe  is  a  discharge  loss  which  should  be  added. 

The  head  may  also  be  obtained  in  a  test  by  taking  the  dif- 
ference between  the  total  heads  (Equation  3)  on  the  suction 
and  discharge  sides  of  the  pump.  If  the  suffix  (S)  signifies  a 
point  in  the  suction  pipe  and  suffix  (D)  a  point  in  the  discharge 
pipe  we  have 

P°        Pa  VD2        TV 


In  this  case  PD/W  represents  the  pressure  gage  reading  reduced 
to  feet  of  water  while  PS/W  represents  the  suction  gage  reading 
reduced  to  feet  of  water.  In  general  the  latter  pressure  will  be 
less  than  that  of  the  atmosphere.  In  such  a  case  ps/w  will  be 
negative  in  value. 

The  word  efficiency  without  any  qualification  will  always 
denote  gross  efficiency,  that  is  the  ratio  of  the  power  delivered 
in  the  water  to  the  power  necessary  to  run  the  pump.  The  hy- 
draulic efficiency  is  the  ratio  of  the  power  delivered  in  the  water 
to  the  power  necessary  to  run  the  pump  after  bearing  friction, 
disk  friction,  and  other  mechanical  losses  are  deducted.  The 
hydraulic  efficiency  is  therefore  equal  to  Wh/Wh"  or  h/h". 
This  latter  expression  is  termed  manometric  efficiency  by  some 
and  is  treated  as  something  essentially  different  from  hydraulic 
efficiency.  If  the  true  value  of  h"  could  be  computed,  the  value 
of  the  hydraulic  efficiency  so  obtained  would  be  the  same  as 
that  obtained  experimentally  by  deducting  mechanical  losses 
from  the  power  necessary  to  drive  the  pump.  ActuallyA-the 
ratio  of  h/h"  will  usually  be  less  than  this  value  but  that  Js  due 
to  the  fact  that  our  theory  is  imperfect.  (Art.  167.) 

163.  Head  Imparted  to  Water.  —  By  reversing  equation  (19) 
in  Art.  60,  since  we  are  now  dealing  with  a'pump  and  not  a  tur- 
bine, we  may  write 

h"  =  7  =  \  (u*Vu*  ~  MlVwi)' 
"k       y 

This  would  be  very  appropriate,  if  the  pump  were  fitted  with 


234  HYDRAULIC  TURBINES 

stationary  guide  vanes  in  the  center  of  the  impeller  to  direct  the 
water  entering.  Occasionally  centrifugal  pumps  are  so  built, 
but  for  the  usual  type  of  pump,  we  may  say  that  whatever  angu- 
lar momentum  the  water  has,  as  it  enters  the  impeller,  it  has 
received  from  the  latter,  through  the  medium  of  intervening 
particles  of  water.  This  is  proven  by  the  fact  that  the  water  in 
the  eye  of  the  impeller  and  even  in  the  suction  pipe  may  be  set 
into  rotation.  The  effect  is  as  if  the  vanes  of  the  impeller, 
extended  to  this  space.  For  this  reason  we  shall  drop  the  last 
term  in  the  above  equation  and  write 

h"  =  -  UtVu,  =  -2  (u2  +  v2  cos  /32)  (76) 

y  y 

By  another  line  of  reasoning,  or  by  a  slight  transformation  .of 
equation  (76),  we  may  obtain 

''-t'-i+g' 

Sometimes  one  of  these  forms  is  more  convenient  than  the  other. 

Inspection  of  equation  (76)  shows  that  if  the  pump  is  to  do 
positive  work,  VuZ  must  be  positive.  Thus  the  absolute  velocity 
of  the  water  must  be  directed  so  as  to  have  a  component  in  the 
direction  of  rotation.  If  the  pump  speed,  w2,  be  assumed  constant, 
equation  (76)  will  plot  as  a  straight  line  for  values  of  vz  (or  q) . 
If  j82  is  less  than  90°,  the  value  of  h"  will  increase  as  the  rate  of 
discharge  increases  above  zero.  If  /32  is  equal  to  90°,  h"  will  be 
independent  of  the  rate  of  discharge  and  will  plot  as  a  horizontal 
line  for  all  values  of  v2.  If  £2  is  greater  than  90°,  the  value  of 
h"  will  decrease  as  the  rate  of  discharge  increases.  (See  Fig.  144.) 

Since  it  is  difficult  to  transform  velocity  head  into  pressure 
head  without  considerable  loss,  it  is  desirable  to  keep  the  abso- 
lute velocity  of  the  water  leaving  the  impeller  as  small  as  pos- 
sible. For  that  reason  the  best  pumps  have  vane  angles  as  near 
180°  as  possible  in  order  that  the  relative  velocity  may  be  nearly 
opposite  to  the  peripheral  velocity  of  the  impeller. 

164.  Losses. — In  accordance  with  the  usual  methods  in 
hydraulics,  the  friction  loss  in  flow  through  the  impeller  may  be 
represented  by  kv22/2g,  where  k  is  an  experimental  constant. 
A  study  of  Fig.  142  would  indicate  that  there  is  no  abrupt  change 
of  velocity  at  entrance  to  the  impeller  under  any  rate  of  flow; 
there  is  then  no  marked  shock  loss  at  entrance  that  would  re- 


CENTRIFUGAL  PUMPS 


235 


quire  the  use  of  a  separate  expression  as  whatever  loss  there  is 
may  be  covered  by  the  value  of  k.  Where  the  water  leaves  the 
impeller,  however,  there  is  an  important  shock  loss  that  follows  a 
different  law  from  the  friction  loss. 

For  the  turbine  pump  this  shock  loss  is  similar  to  that  in  the 
case  of  the  reaction  turbine  in  Art.  86.     Referring  to  Fig.  143,  it 


FIG.  142. — Velocity  diagram  for  three  rates  of  discharge. 

may  be  seen  that  the  velocity  F2  and  the  angle  az  will  be  deter- 
mined by  the  vectors  HZ  and  v2.  Since  the  vane  angle  a' 2  is 
fixed  there  can  be  only  one  value  of  the  discharge  that  does  not 
involve  a  shock  loss.  For  any  other  value  of  the  discharge  the 
velocity  F2  will  be  forced  to  become  V'%  with  a  resultant  loss 


236 


HYDRAULIC  TURBINES 


which  may  be  represented  by  (CC')2/2g.  Since  the  area  of  the 
diffusion  ring  normal  to  the  radius  should  equal  the  area  of  the 
impeller  normal  to  the  radius,  the  normal  component  (i.e.,  per- 
pendicular to  u2)  of  V2  should  equal  that  of  V*.  Therefore  CC' 
is  parallel  to  u2  and  its  value  may  be  found  to  be 


=  U2 


If  A;' 


sn 


-a'8) 


sn  ft  2  -  < 

•—•  -  / 
sm  a'2 


sin  a  2 
is  approximately  equal  to 


then  for  the  turbine  pump  the  shock  loss 


(u2  - 


For  the  turbine  pump  the  total  hydraulic  loss  may  be  repre- 
sented by 

7   /  7      ^2  i         V^2  *C    ^2/  /^TO\ 

ft   =fc2^+       ^7"  (78) 


Since  the  volute  pump  has  no  diffusion  vanes,  there  will  be  no 
abrupt  change  in  the  direction  of  the  water  at  exit  from  the  im- 
peller, but  there  may  be  an  abrupt  change  in  the  magnitude  of 
the  velocity.  The  water  leaves  the  impeller  with  a  velocity  V2 
and  enters  the  body  of  water  in  the  case  which  is  moving  with  a 
velocity  V$.  In  accordance  with  the  usual  law  in  hydraulics 
this  shock  loss  may  be  represented  by 


20 

For  the  usual  type  of  pump  F2  will  decrease  as  the  discharge  in- 
creases, and  in  any  case  F3  must  increase  as  the  quantity  of 
water  becomes  greater.  If  the  discharge  becomes  such  that  the 
two  are  equal  then  there  will  be  no  shock  loss.  The  value  of  V2 
may  be  expressed  in  terms  of  u2  and  v2,  and  if  the  ratio  of  (a2/A3) 
be  denoted  by  n,  we  have  V$  =  nv2.  Making  these  substitutions 

*L.  M.  Hoskins,  "Hydraulics,"  p.  237. 


CENTRIFUGAL  PUMPS  237 

the  total  hydraulic  loss  for  the  volute  pump  may  be  represented 
by 


,  ,  _  ,         ,  2  +  2u2v2  cos 


Though  the  values  of  k  may  be  different  for  the  two  types  and 
though  the  expressions  for  shock  loss  are  unlike  in  appearance, 
yet  it  can  be  seen  that  the  losses  in  each  case  follow  the  same 
general  kind  of  a  law.  In  the  turbine  pump  we  have  a  gradual 
reduction  of  velocity  but,  except  for  one  value  of  discharge,  a 
sudden  change  in  direction  as  the  water  leaves  the  impeller. 
With  the  volute  pump  we  have  no  abrupt  change  of  direction  but; 
a  sudden  change  of  velocity.  The  transformation  of  kinetic 
energy  into  pressure  energy  is  incomplete  in  either  case,  but  it 
is  generally  believed  that  the  loss  is  somewhat  greater  in  the 
volute  pump  than  in  the  turbine  pump. 

For  an  infinitesimal  discharge  the  value  of  the  velocity  in  the 
case,  Vs,  would  be  practically  zero.  Therefore  a  particle  of 
water  leaving  the  impeller  with  a  velocity  V2  and  entering  a  body 
of  water  at  rest  would  lose  all  its  kinetic  energy.  For  such  a 
case,  however,  the  value  of  v2  would  be  also  practically  zero  so 
that  V2  would  equal  u2.  Therefore  for  a  very  slight  discharge  the 
shock  loss  would  be  hf  =  u22/2g.  Such  a  value  of  h'  may  be  ob- 
tained from  either  (78)  or  (79)  by  putting  v2  =  0. 

165.  Head  of  Impending  Delivery.  —  The  head  developed  by 
the  pump  when  no  flow  occurs  is  called  the  shut-off  head  or  the 
head  of  impending  delivery.  We  are  then  concerned  only  with 
the  centrifugal  head  or  the  height  of  a  column  of  water  sustained 
by  centrifugal  force.  In  Art.  160  this  was  shown  to  be  equal  to 
u22/2g.  The  same  result  may  be  obtained  from  the  principles 
of  Art.  163  and  Art.  164.  If  v2  becomes  zero,  then  by  equation 
(76),  h"  =  u^/g  =  2  u22/2g.  But,  as  was  shown  in  Art.  164,  the 
loss  of  head,  h'  =  u22/2g.  Therefore  h  =  h"  -  In!  =  u22/2g. 

Although  ideally  the  head  of  impending  delivery  equals  u22/2g, 
we  find  that  various  pumps  give  values  either  above  or  below 
that.  This  may  be  accounted  for  in  a  number  of  ways.  In  any 
pump  we  never  have  a  real  case  of  zero  discharge;  for  a  small 
amount  of  water,  about  5  per  cent,  of  the  total  rated  capacity 
perhaps,  will  be  short  circuited  through  the  clearance  spaces.  A 
pump  is  said  to  have  a  rising  characteristic  if,  when  run  at  con- 
stant speed,  the  head  increases  as  the  discharge  increases  above 
zero  until  a  certain  value  is  reached  and  then  begins  to  decrease. 


238  HYDRAULIC  TURBINES 

If  the  head  continually  decreases  as  the  discharge  increases  above 
zero,  the  pump  is  said  to  have  a  falling  characteristic.  Thus  the 
leakage  through  the  clearance  spaces  will  tend  to  make  the  shut- 
off  head  greater  or  less  than  u22/2g  according  to  whether  the  pump 
has  a  rising  or  a  falling  characteristic.  The  more  the  vanes  are 
directed  backward,  the  more  tendency  there  is  for  internal  eddies 
to  be  set  up  and  these  tend  to  decrease  the  head.  Also  if  the 
water  in  the  eye  of  the  impeller  is  not  set  in  rotation  at  the  same 
speed  as  the  impeller  the  head  may  be  further  reduced.  There 
is  also  a  tendency  for  the  water  surrounding  the  impeller  to  be 
set  in  rotation  but  this,  on  the  other  hand,  helps  to  increase  the 
head  since  the  real  value  of  rz  is  greater  than  the  nominal  value. 

It  will  usually  be  found  that  actually  the  head  of  impending 

u  2 
delivery  may  be  from  0.9  to  1.1  -~-" 

166.  Relation  between  Head,  Speed  and  Discharge. — When 
flow  occurs  the  above  relation  no  longer  holds,  for  other  factors 
besides  centrifugal  force  enter  in.  Due  to  conversion  of  velocity 
head  into  pressure  head  when  water  flows,  a  lift  may  be  obtained 
which  is  greater  than  u22/2g.  (See  Fig.  144.) 

This  may  be  shown  best  by  equation  (77),  when  the  losses  are 
introduced.  The  hydraulic  friction  loss  in  flow  through  the  im- 
peller may  be  represented  by  kv22/2g.  Then  at  discharge  from 
the  impeller  a  portion  of  the  kinetic  energy  is  lost  within  the 
diffuser  or  within  the  volute  case,  and  the  remainder  may  be 
represented  by  mV22/2g,  where  m  is  a  factor  less  than  unity. 
Deducting  the  losses  from  the  expression  for  h"  in  equation  (77) 
we  have 

5;|W;'^>||*W  (80) 

If  the  factor  involving  V2  is  greater  than  that  with  v2  the  head  will 
be  greater  than  the  shut-off  head,  while  the  reverse  is  true  if  it 
is  less. 

In  order  to  produce  a  pump  with  a  rising  characteristic,  it  is 
not  only  necessary  to  conserve  the  kinetic  energy  of  the  water 
discharged  from  the  impeller,  or  in  other  words  to  keep  the  factor 
m  high,  but  it  is  also  necessary  to  have  V2  large  and  v2  small. 
But  a  pump  with  a  falling  characteristic  is  not  necessarily  any  less 
efficient  than  the  former  type.  The  factor  m  may  be  high  but  yet 
V2  may  be  made  low  and  v2  high.  In  order  to  accomplish  these 
results,  it  may  be  seen  that  the  vane  angle  /32  has  some  influence, 


CENTRIFUGAL  PUMPS 


239 


but.it  is  not  the  sole  determining  factor  that  many  have  supposed. 
It  must  be  noted  that  we  are  concerned  with  h,  which  is  a  very 
different  quantity  from  h",  and  hence  the  remarks  in  Art.  163 
cannot  be  applied  here.  In  fact  it  has  been  shown  that  pumps 
with  vane  angles  greater  than  90,°  and  in  fact  as  large  as  154,° 


Discharge 
FIG.  144. —  Ideal  curves  for  a  turbine  punp. 

may  manifest  decidedly  rising  characteristics,  while  certain  im- 
pellers with  radial  vanes  have  given  steep  falling  characteristics 
and  not  flat  characteristics.1 

While  the  above  equation  is  very  satisfactory  in  explaining 
how  the  value  of  h  may  either  increase  or  decrease,  and  in  comput- 

J  See  the  author'^"  Centrifugal  Pumps." 


240  HYDRAULIC  TURBINES 

ing  its  value  for  some  specified  condition,  such  as  that  for  maxi- 
mum efficiency,  it  is  not  the  best  form  of  equation  for  showing  the 
complete  characteristic.  This  is  largely  due  to  the  fact  that  the 
factor  m  is  a  variable,  ranging  all  the  way  from  zero  up  to  a 
maximum  of  about  0.75  at  the  condition  for  highest  efficiency, 
and  also  the  rate  of  discharge  affects  two  different  variables  v2 
and  Vz,  which  are  in  reality  related  to  each  other.  For  some 
purposes,  therefore,  it  is  better  to  derive  the  following  forms 
of  equations. 

The  actual  lift  of  the  pump  h  may  be  obtained  by  subtracting 
the  losses  hr  from  the  head  h"  imparted  by  the  impeller.     The 

,  ,„     .„  i  u2  (uz  +  v%  cos  02)  ,         c  ,  , 

value  of  h    will  be  taken  as  —  -  and  the  values  of  h 

are  given  in  equations  (78)  and  (79). 

Making  these  substitutions  for  the  turbine  pump  we  obtain 
after  reduction 


U22  +  2  (kf  +  cos/32)  u2v2  -  (k  +  &'2)  vz2  =  2gh.         (81) 
For  the  volute  pump  we  obtain  after  rearranging 


2nv2  Vw22      2uzv2  cos 


These  equations  involve  the  relation  between  the  three  vari- 
ables u2,  vz,  and  h.  Any  one  of  these  may  be  taken  as  constant 
and  the  curve  for  the  other  two  plotted.  If  the  pump  is  to  run 
at  various  speeds  under  a  constant  head,  the  latter  will  then  be 
fixed  and  we  may  determine  the  relation  between  speed  and 
discharge.  The  more  common  case  is  for  the  pump  to  run  at  a 
constant  speed.  For  that  case  values  of  h  may  be  computed  for 
different  values  of  v2.  The  curves  for  a  turbine  pump  run  at 
constant  speed  are  shown  in  Fig.  144. 

Although  it  will  not  be  done  here,  it  will  be  found  convenient 
to  introduce  ratios  or  factors  as  was  done  in  the  case  of  the  tur- 
bine. We  may  write  uz  =  <t>\/2gh  and  v2  =  c\/2gh  and  using 
these  in  equations  (81)  and  (82)  we  obtain  relations  between  c 
and  (f>  similar  to  equation  (40).  As  in  the  case  of  the  turbine 
it  will  be  found  that  the  best  efficiency  will  be  obtained  for  a 
certain  value  of  <f>  and  c.  It  will  thus  be  clear  that  the  speed  of 
the  pump  should  vary  as  the  square  root  of  the  lift,  and  that  the 
best,  value  of  the  discharge  will  be  proportional  to  the  square 

root  of  the  lift.     Since  h  =  Tii  it  is  apparent  that  the  lift 


CENTRIFUGAL  PUMPS 


241 


varies  as  the  square  of  the  speed.     If  this  value  of  h  be  substi- 
tuted in  v2  =  c\/2gh  we  obtain  v 2  =  I— J  uz,  and  this  shows  that  the 

best  value  of  the  discharge  varies  directly  as-  the  speed. 

Curves  between  c  and  </>  will  be  of  the  same  appearance  as  those 
drawn  for  a  constant  value  of  h.  To  construct  curves  of  the  same 
shape  as  those  drawn  for  a  constant  speed  it  will  be  necessary  to 

plot  values  of  (—^  and  of  I— )  • 


Discharge 
FIG.  145. —  Actual  curves  for  turbine  pump. 

The  value  of  4>  for  the  maximum  efficiency  depends  upon  the 
design  of  the  pump.  By  choosing  different  values  of  @2  and  either 
a  '2  or  n,  and  different  numbers  of  impeller  vanes  and  other  fac- 
tors, a  pump  may  be  given  a  rising  or  a  flat  or  a  steep  falling 
characteristic.  The  values  of  <f>  for  the  highest  efficiency  range 
from  about  1.30  down  to  about  0.90.  This  means  that  the 
normal  head  is  usually 

h  =  0.6  to  l.l^2- 

The  value  of  cr,  the  coefficient  of  the  radial  velocity  at  the  point 
of  outflow  from  the  impeller,  is  usually  from  0.05  to  0.15.  All 
formulas  and  values  in  this  chapter  are  based  upon  the 
head  developed  per  stage. 

16 


242 


HYDRAULIC  TURBINES 


167.  Defects  of  Theory. — The  discussion  of  the  defects  of 
the  theory  of  the  reaction  turbine  in  Art.  92  applies  equally 
well  to  the  centrifugal  pump.  Probably  one  of  the  greatest 
sources  of  error  lies  in  the  assumptions  made  regarding  losses. 
In  particular  the  expressions  for  shock  loss  for  either  the  tur- 
bine or  the  volute  pump  must  be  regarded  as  only  rough 
approximations. 


Discharge 
FIG.  146. — Curves  for  pump  run  at  different  speeds. 

While  actual  tests  have  shown  curves  similar  to  the  ideal 
curves  given  in  Fig.  144,  it  is  more  common  to  find  the  relation 
between  head  and  discharge  to  be  like  that  in  Fig.  145.  In  many 
cases  also  the  pump  has  a  falling  characteristic  so  that  the  head 
for  any  delivery  is  less  than  the  shut-off  head.  But  at  the  same 
time  the  gross  efficiency  will  be  high  and  the  hydraulic  efficiency 
must  be  still  higher.  Since  h"  =  h/e  it  will  be  seen  that  the 


CENTRIFUGAL  PUMPS  243 

actual  h"  must  be  of  the  form  shown  in  Fig.  145.  This  accounts 
for  the  discrepancy  between  the  so-called  manometric  efficiency 
and  the  true  hydraulic  efficiency. 

The  reasons  for  this  are  the  same  as  those  given  for  the  reaction 
turbine.  In  addition  there  is  strong  reason  for  believing  that 
there  is  a  dead  water  space  on  the  rear  of  each  vane,  thus  the 
actual  area  a2  will  be  less  than  the  nominal  area  used  in  the 
computations.  This  is  probably  a  larger  item  than  the  contrac- 
tion of  the  streams  mentioned  in  connection  with  the  turbine. 
The  ordinary  pump  has  no  guide  vanes  at  entrance  to  the  impeller 
and  the  conditions  of  flow  at  that  point  are  uncertain. 

The  more  vanes  the  impeller  has  the  more  perfectly  the  water 
is  guided  and  the  more  nearly  the  actual  curves  approach  the 
ideal.  It  is  necessary  to  have  enough  vanes  to  guide  the  water 
fairly  well  but  too  many  of  them  cause  an  excessive  amount  of 
hydraulic  friction.  Within  reasonable  limits — say  6  to  24 — the 
efficiency  is  but  little  affected.  If  the  use  of  few  vanes  lowers 
the  value  of  h,  the  value  of  h"  is  lowered  at  about  the  same  rate 
so  that  the  ratio  of  the  two  is  but  little  altered. 

168.  Efficiency  of  a  Given  Pump. — If  a  given  pump  is  run 
at  different  speeds  the  lift  should  vary  as  the  square  of  the  speed, 
the  discharge  as  the  speed,  and  the  water  h.p.  as  the  cube  of 
the  speed.  If  the  efficiency  of  the  pump  remained  constant  the 
horsepower  necessary  to  run  the  pump  would  also  vary  as  the 
cube  of  the  speed.  It  is  probable  that  the  hydraulic  efficiency 
is  reasonably  independent  of  the  speed.  The  mechanical  losses, 
however,  do  not  vary  as  the  cube  of  the  speed.  For  low  speeds 
the  mechanical  losses  do  not  increase  so  fast  and  thus  the  gross 
efficiency  of  the  pump  will  improve  as  it  is  used  under  higher 
heads  at  higher  speeds.  After  a  certain  limit  is  reached,  how- 
ever, the  mechanical  losses  follow  another  law  and  for  very  high 
speeds  they  will  increase  faster  than  the  hydraulic  losses  and 
the  efficiency  will  begin  to  decline.  Thus  for  a  given. pump 
run  at  increasing  speeds  the  maximum  efficiency  will  increase 
and  then  decrease  again.  It  is  thus  clear  that  the  head  which 
may  be  efficiently  developed  with  a  single  stage  is  limited. 
For  higher  heads  it  is  necessary  to  resort  to  multi-stages. 

These  conclusions  regarding  efficiency  are  borne  out  by  the 
curves  shown  inJFig.  146  and  Fig.  147.  In  the  latter  the  highest 
speed  attained  was  not  sufficient  for  the  efficiency  to  begin  to 
decrease* again,  though  it  had  evidently  reached  its  limit. 


244 


HYDRAULIC  TURBINES 


In  a  set  of  curves  such  as  are  given  in  Fig.  146,  the  operation 
of  the  pump  under  a  constant  head  can  be  determined  by  fol- 
lowing a  horizontal  line.  For  a  constant  discharge  follow  a  ver- 
tical line,  and  to  determine  the  conditions  for  maximum  efficiency 
follow  the  curved  dotted  line.  The  values  of  the  maximum 
efficiency  will  be  given  by  the  curve  tangent  to  the  peaks  of  all 
the  efficiency  curves  for  the  various  speeds. 


900     1000    1100  .   1200     1300     1400     1500     1600    1700 

R.P.M. 

FIG.  147. — Characteristic  curve  of  a  4-stage  turbine  pump. 


1800 


169.  Efficiency  of  Series  of  Pumps. — For  a  given  pump  the 
speed  and  head  are  seen  to  have  some  influence  upon  its  efficiency. 
However,  the  capacity  for  which  it  is  designed  is  the  greatest 
factor.  Suppose  we  have  a  series  of  impellers  of  the  same 
diameter  and  same  angles  running  at  the  same  speed,  the  lift 
will  be  approximately  the  same  for  all  of  them.  Suppose,  how- 
ever, that  the  impellers  are  of  different  widths.  The  discharge 
will  then  be  proportional  to  the  width  and  the  water  horsepower 
is  proportional  to  the  discharge.  But  the  bearing  friction  and  the 
disk  friction  are  practically  the  same  for  all  of  them.  In  addition 
the  hydraulic  friction  in  the  narrow  impellers  will  be  greater 
than  that  in  the  larger  ones.  It  is  therefore  evident  that  the 
efficiency  of  the  high-capacity  impellers  will  be  much  greater 
than  that  of  the  low-capacity  impellers.  This  is  true  to  such 


CENTRIFUGAL  PUMPS 


245 


an  extent  that  the  efficiency  of  a  centrifugal  pump  may  be  said 
to  be  a  function  of  the  capacity.  (See  Fig.  148.) 

Very  large  pumps  have  given  efficiencies  around  90  per  cent. 
Single  suction  pumps  have  slightly  lower  efficiencies  than  double 
suction  pumps. 

170.  Specific  Speed  of  Centrifugal  Pumps. — The  specific  speed 
factor  for  centrifugal  pumps  is  as  useful  as  that  for  hydraulic 
turbines.  By  it,  we  can  at  once  determine  the  conditions  that 
are  possible  for  a  pump  of  existing  design,  and  can  also  select  the 
most  suitable  combination  of  factors  for  a  proposed  pump  for 
any  case.  It  also  serves  to  classify  pump  impellers  as  to  type 
in  the  same  way  that  it  indicates  the  type  of  turbine  runner. 


100 
90 

I   8° 
£   70 

I   60 
I 
W   50 

40 
30 


2,000 


10,000 


4,000  6,000  8,000 

Capacity  in  Gal.  per  Min. 
FIG.  148. — Efficiency  as  a  function  of  capacity, 


12,000 


14,000 


Thus  a  low  value  of  the  specific  speed  indicates  a  narrow  impeller 
of  large  diameter,  while  the  reverse  is  true  for  a  high  value. 

For  the  centrifugal  pump,  however,  it  is  more  convenient  to 
use  a  different  form  for  this  factor  than  for  the  turbine.  We 
are  not  primarily  concerned  with  the  power  required  to  drive 
a  pump,  but  have  our  attention  centered  first  upon  its  capacity. 
But  since  the  capacity  and  power  are  directly  related,  it  is  seen 
that  we  are  merely  expressing  the  specific  speed  in  different 
units.  Since,  as  in  the  case  of  the  turbine,  q  =  KiD^^/h,  and 
N  =  18400  \/h/D,  we  may  eliminate  D  between  the  two  equa- 
tions and  obtain 

•     -N't  =  18400  V^i  = 


246 


HYDRAULIC  TURBINES 


But  centrifugal  pumps  are  usually  rated  in  gallons  per  minute 
rather  than  in  cubic  feet  per  second  and  it  may  be  more  con- 
venient to  express  the  above  as 


N.  = 


N\/G.P.M. 

pi 


(84) 


Since  448  G.P.M.  =  1  cu.  ft.  per  second,  it  may  be  seen  that 
N.  =  2l.2N'8. 

For  a  single  impeller,  values  of  Na  ordinarily  range  from  500 
to  8000.  This  latter  figure  has  been  greatly  exceeded  in  a  few 
cases  of  special  types.  Just  as  in  the  case  of  the  turbine,  the 
efficiency  may  be  expressed  as  a  function  of  the  specific  speed, 
as  is  shown  in  Fig.  149. 


100 


80 


-60 


50 


1,000 


2,000  3,000  4,000 

Specific  Speed,  Ns   -N)^ 


5,000 


6,000 


7,000 


FIG.  149. — Efficiency  as  a  function  of  specific  speed. 

171.  Conditions  of  Service. — Centrifugal  pumps  are  used  for 
lifting  water  to  all  heights  from  a  few  feet  to  several  thousand. 
Several  pumps  have  been  built  to  work  against  a  head  of  2000 
ft.,  though  these  are  all  multi-stage  pumps.  The  usual  head  per 
stage  is  not  more  than  100  to  200  ft.,  though  this  figure  has  been 
exceeded  in. numerous  instances. 

The  capacities  of  centrifugal  pumps  ranges  all  the  way  from 
very  small  values  up  to  300  cu.  ft.  per  second  or  134,500  gal.  per 
minute.  Rotative  speeds  range  ordinarily  from  30  to  3000 
r.p.m.  according  to  circumstances.  All  the  above  figures  are  for 
ordinary  practice,  and  are  not  meant  to  be  the  limiting  values 
that  can  be  used. 


CENTRIFUGAL  PUMPS  247 

172.  Construction. — The  design  and  construction  of  centrifugal 
pumps  is  very  similar  in  principle  to  that  of  reaction  turbines. 
Impellers  are  made  either  single  suction  or  double  suction,  ac- 
cording to  whether  water  is  admitted  at  only  one  or  both  sides 
of  the  impeller.  The  latter  construction  permits  of  a  smaller 
diameter  of  impeller  for  the  same  capacity. 

Water  leakage  from  the  discharge  to  the  suction  side  is  minim- 
ized by  the  use  of  clearance  rings,  as  in  the  case  of  turbines,  and 
sometimes  labyrinth  rings  are  used  so  as  to  provide  a  more  tor- 
tuous passage  for  the  leakage  water.  The  leakage  of  air  along 
the  shaft  on  the  suction  side  should  be  prevented  by  a  water  seal 
in  addition  to  the  usual  packing. 

The  end  thrust  is  taken  care  of  by  a  thrust  bearing,  by  sym- 
metrical construction,  as  in  the  case  of  the  double  suction  pump  or 
a  multi-stage  pump  with  impellers  set  back  to  back,  or  by  use  of 
an  automatic  hydraulic  balancing  piston.  The  majority  of  multi- 
stage pumps  are  built  with  the  impellers  all  arranged  the  same 
way  in  the  case  as  this  permits  the  most  direct  flow  from  one 
impeller  to  the  next  and  also  simplifies  the  mechanical 
construction. 

173.  QUESTIONS  AND  PROBLEMS 

1.  Why  is  the  centrifugal  pump  so  called?     How  does  the  pressure  and 
velocity  of  the  water  vary  as  it  flows  through  such  a  pump? 

2.  What  classes  of  centrifugal  pumps  are  there,  and  how  do  they  differ? 

3.  What  is  the  difference  between  the  head  imparted  to  the  water  and 
the  head  developed  by  the  pump?     How  is  the  latter  measured  in  a  test? 

4.  What  are  the  important  hydraulic  losses  in  the  centrifugal  pump? 
What  is  meant  by  the  head  of  impending  delivery?     What  is  its  approximate 
value? 

5.  How  may  the  head  vary  with  the  rate  of  discharge,  the  speed  being 
constant?     Why  is  this? 

6.  If  a  given  centrifugal  pump  is  run  at  a  different  speed  how  will  the 
head,  rate  of  discharge,  power,  and  efficiency  vary,  assuming  that  the  con- 
ditions are  such  that  <j>  is  constant? 

7.  Why  is  the  efficiency  of  a  centrigufal  pump  a  function  of  its  capacity? 

8.  What  is  meant  by  the  specific  speed  of  a  centrifugal  pump?     What 
is  the  use  of  such  a  factor? 

9.  The  diameter  of  the  impeller  of  a  single-stage  centrifugal  pump  is  6  in. 
If  it  runs  at  2000  r.p.m.,  what  will  be  the  approximate  value  of  the  shut-off 
head  and  the  head  for  the  rate  of  discharge  corresponding  to  maximum 
efficiency?  Ans.     42.5  ft.  and  30  ft. 

10.  What  may  be  the  maximum  and  minimum  limits  of  the  capacity  of 
a  series  of  pumps  of  the  same  diameter  as  the  single  pump  in  problem  (9) 
and  running  at  the  same  speed  ? 


248  HYDRAULIC  TURBINES 

11.  What  will  be  the  answers  to  problems  (9)  and  (10)  if  the  pump  is  a 
four-stage  pump,  all  other  data  remaining  the  same? 

12.  If  a  single-stage  centrifugal  pump  is  to  develop  a  shut-off  head  of  200 
ft.,  what  must  be  its  r.p.m.,  if  the  impeller  diameter  is  18  in.? 

Ans.     1446  r.p.m. 

13.  If  the  above  pump  were  a  two-stage  unit,  what  would  be  the  necessary 
speed?  Ans.     1045  r.p.m. 

14.  Compute  the  specific  speed  for  the  pumps  in  problems  (12)  and  (13). 

15.  Compute  the  factors  by  which  u<?/2g  must  be  multiplied  to  give  the 
shut-off  head  and  the  head  for  highest  efficiency  for  the  pumps  whose  tests 
are  given  in  Appendix  C,  Tables  14  and  15. 


APPENDIX  A 
THE  RETARDATION  CURVE 

Let  the  relation  between  instantaneous  speed  and  time  be 
represented  by  the  curve  shown  in  the  figure.    Let 
N  =  r.p.m. 
t     =  seconds. 

s     =  length  of  subnormals  in  inches. 
x    =  distance  in  inches. 
y    =  distance  in  inches. 
m  =  seconds  per  inch. 
n    =  r.p.m.  per  inch. 

/    =  moment  of  inertia  of  the  rotating  mass  in  ft.-lb.  sec.2 
units. 


FIG.  150. 

Thus  y  =  N/n  and  x  =  t/m 

co  =  radians  per  second  =  2irN/60.     da>/dt  =  (2<,r/6Q)dN/dt. 

From  mechanics,  Torque  =  Idu/dt.     Power  =  Iwdw/dt. 

Power  =  (2ir/wyiNdN/dt 

Tan  0  =  dy/dx  =  (dN/n)  +  (dt/m). 

But  also  tan  <£  =  s/y.     Equating  these  two,  dN/dt  =  ns/my. 
Thus 

Power  =  (27T/60)2  (n2/m)Is 
=  Ks 
249 


250  HYDRAULIC  TURBINES 

This  gives  the  value  of  power  in  ft.-lb.  per  second.  To  obtain 
horse-power  or  kilowatts  it  is  necessary  to  introduce  the  proper 
constants  in  computing  K.  If  the  moment  of  inertia  could  be 
computed  or  determined  experimentally  the  value  of  K  could  be 
obtained  from  the  above.  Usually  it  is  necessary  to  obtain  K 
by  direct  experiment. 


APPENDIX  B 


STREAM  LINES  IN  CURVED  CHANNELS 

The  following  theory  is  based  upon  certain  assumptions  which 
are  only  approximately  realized  in  practice,  but  yet  there  are 
many  cases  which  approach  these  conditions  so  closely  that  the 
methods  here  given  may  be  successfully  applied.1  Assume  that 
across  any  section,  such  as  AB  in  Fig.  151,  the  total  head  is 
constant.  This  will  be  true  if  all  particles  of  water,  coming  from 
some  source,  have  lost  equal  amounts  of  energy  en  route  and  thus 
all  reach  the  section  AB  with  an  equal  store  of  energy.  Actually 
some  particles  of  water  may  have  lost  more  than  others.  But 


FIG.  151. 

if- it  be  assumed  that  the  total  head  across  the  section  is  constant, 
it  follows  that,  if  the  pressure  is  higher  at  any  point,  the  velocity 
will  be  lower  than  at  some  other  point  and  vice  versa.  Owing  to 
centrifugal  action,  the  pressure  at  B  will  be  greater  than  that  at 
A  and  hence  the  velocity  will  decrease  along  the  line  from  A  to 
5,  the  line  AB  being  normal  to  the  stream  lines. 

In  Fig.  151,  let  us  consider  an  elementary  volume  of  water 


1  From  notes  by  Lewis  F.  Moody. 


251 


252  HYDRAULIC  TURBINES 

whose  length  in  the  direction  AB  is  dn,  the  area  of  the  face  per- 
pendicular to  AB  being  AA.  The  mass  is  then  wAAdn/g.  Let 
the  stream  line  in  question  be  at  a  distance  n  from  the  wall  at 
A  and  have  a  radius  of  curvature  of  p.  Considering  forces  along 
the  normal  line  AB,  we  have  dp  X  A  A  due  to  the  difference  in 
pressure  on  the  two  forces  and  wAAdn  cos  a  as  the  component 
of  gravity.  The  normal  acceleration  is  F2/P-  Hence  we  may 
apply  the  proposition  that  force  equals  mass  times  acceleration 
and  obtain 

dp&A  +  wAAdn  cos  a  =  (w&Adn/g)(V2/p) 

Letting  dz  represent  the  change  in  elevation  corresponding  to 
dn,  we  have  dz  =  dn  cos  a.  Thus  from  the  above  we  may  write 

gdp   ,gdz=V^  _ 

wdn      wdn       p 

V2 

-^-  =  constant 

differentiate  with  respect  to  n  and  obtain 

dH  =    dp 
dn       wdn 

And  from  this  we  may  write 

(86) 


p  V2 

Since  H  =  -  +  z+-^-  =  constant  along  line  A  B}  we  may 


dH  =    dp         dz       2VdV  = 
dn       wdn       dn        2gdn 


wdn        dn          dn 
Combining  equations  (85)  and  (86)  we  obtain 


P  dn 

This  may  be  written  as 

dV        _dn 
V .'       '  p 

Integrating 

V 


V^ 

VA  = 

V  =  ~^rdn  =  ^  (87) 

+  I  e 

e    J°    p 

;  ( »     ' 

In  the  above,   VA  is  the  velocity  next  to  the  wall  at  A,  where 
n  =  0. 


I        STREAM  LINES  IN  CURVED  CHANNELS       253 

In  general,  sketch  in  tentative  flow  lines.  Measure  the  curva- 
ture and  plot  values  of  1/p  as  a  function  of  n.  The  area  between 
the  curve  and  the  n  axis  is  the  value  of  the  integral.  Denote  this 
area  by  Yi.  Let  F2  =  l/eY\  Then  V  =  VAY2.  Hence  if  the 
velocity  at  A  were  known,  the  velocity  along  any  other  stream 
line  crossing  AB  could  be  determined. 

The  solution  of  the  problem  from  this  point  depends  upon  the 
variation  in  the  cross-section  perpendicular  to  the  plane  of  the 
paper.  The  remainder  of  the  discussion  will  be  confined  to  the 
case  where  the  boundary  walls  are  planes  passing  through  an 
axis  of  rotation,  as  shown.  The  thickness  of  the  elementary 


Values  of  n 
FlG.   152. 

volume  perpendicular  to  the  plane  of  the  paper  will  be  rA0. 
The  rate  of  discharge  through  the  channel  between  the  wall  at 
A  and  the  stream  line  at  distance  n  will  be 

\  0  .  dn  .  V. 

For  the  entire  circumference  around  the  axis  from  which  r  is 
measured,  we  may  substitute  2x  for  A0,  and,  inserting  the  value 
of  V  given  by  equation  (87),  we  have 


Y3dn 


(88) 


In  order  to  evaluate  this,  plot  a  curve  for  values  of  r/eYl  =  r 
YZ  =  YZ  as  a  function  of  n.     Denote  the  area  under  this  curve 


254 


HYDRAULIC  TURBINES 


by  F4.  The  ordinates  of  the  curve  to  F4  in  Fig.  152  indicate 
values  of  qf  from  the  wall  at  A  up  to  any  stream  line.  When 
n  =  AB,  the  value  of  q'  =  q,  which  is  the  known  rate  of  discharge 
through  the  entire  channel.  This  final  ordinate  may  then  be 
divided  up  into  any  number  of  equal  parts  desired  and  corre- 
sponding values  of  n  determined  from  the  F4  curve.  This  fixes 
the  division  points  for  the  stream  lines  along  the  section  AB. 
A  similar  procedure  may  be  gone  through  with  for  any  other 
sections.  If  a  considerable  change  is  effected  in  the  tentative 
stream  lines  first  assumed,  this  may  be  repeated  for  the  corrected 


FIG.  153. 

set  and  so  on.     However  extreme  accuracy  is  not  warranted,  so 
that  a  reasonable  approximation  is  quite  sufficient. 

As  a  final  check  on  the  work  it  may  be  noted  that  if  a  series 
of  normal  lines  be  drawn,  as  in  Fig.  153, 

An  X  r 


As 


=  constant 


(89) 


The  proof  of  this  is  that  if  two  flow  lines  are  spaced  dn  apart,  we 
may  write  Asi/As2  =  P/(P  +  dn).  From  the  preceding  treat- 
ment, we  have  dV/V  —  —  dn/p,  from  which  may  be  obtained 
(V  +  dV)/V  =  Vi/Vt=  (P  -  dn)/p.  Multiplying  both  num- 
erator and  denominator  of  the  last  term  by  (p  -\-  dn)  and  drop- 
ping differentials  of  the  second  order,  we  have 


or  FAs  =  constant 


(90) 


This  shows  that  the  velocities  along  any  normal  line  are  inversely 


STREAM  LINES  IN  CURVED  CHANNELS       255 

proportional  to  the  distances  between  the  successive  normal 
lines. 

If  the  distance  between  two  bounding  surfaces,  measured 
perpendicular  to  the  plane  of  the  paper  is  rA0,  we  may  write 

a'  —  AnrA0F  =  AnrA0  X  -          — ,  since  VAs  =  constant. 

As 

If  all  stream  lines  are  so  spaced  as  to  subdivide  the  total  flow 
into  equal  parts,  we  have  for  the  entire  channel  Anr/As  = 
constant. 

In  the  profile  views  of  the  turbine  runner  are  shown  only  the 
circular  projections  of  the  true  stream  lines.  The  application  of 
the  preceding  theory  to  this  case  is  open  to  some  uncertainty, 
but  the  theory  should  apply  rather  closely  to  the  stream  lines  from 
the  draft  tube  up  to  the  discharge  edge  of  the  runner,  since  these 
lines  should  be  in  the  plane  of  the  paper.  The  principal  object 
of  the  procedure  is  to  determine  the  division  points  along  the 
outflow  edge  and  the  direction  of  the  stream  lines  at  these  points, 
and  any  uncertainty  as  to  the  stream  lines  within  the  runner  will 
have  little  effect  upon  this.  Hence  the  method  is  acceptable. 


APPENDIX  C 
TEST  DATA 

The  following  data  will  supply  material  from  which  a  number 
of  curves  may  be  constructed.  Most  of  it  will  be  found  suitable 
for  plotting  characteristic  curves,  if  desired. 

Tables  1  to  5  inclusive  are  Holyoke  tests  ofcfive  reaction  tur- 
bines of  different  types,  taken  from  "  Characteristics  of  Modern 
Hydraulic  Turbines"  by  C.  W.  Lamer  in  Trans.  A.  S.  C.  E.,  Vol. 
LXVI,  p.  306.  Table  6  contains  the  results  of  the  test  of  an 
I.  P.  Morris  turbine  in  the  hydro-electric  plant  of  Cornell  Uni- 
versity. 

Tables  7  to  11  inclusive  are  tests  of  the  same  Pelton-Doble 
tangential  water  wheel  under  widely  different  heads.  These 
were  made  under  the  direction  of  the  author  by  F.  W.  Hoyt  and 
H.  H.  Elmendorf,  seniors  in  Sibley  College.  In  general  they  con- 
firm the  conclusions  in  Art.  103.  Within  reasonable  limits  the 
characteristic  curve  is  about  the  same  regardless  of  the  head 
under  which  the  test  was  made.  The  results  show  that  the 
efficiency  increases  rather  rapidly  as  the  head  is  increased  from 
very  low  values,  but,  as  the  effect  of  mechanical  losses  becomes 
relatively  less  for  the  higher  heads,  the  efficiency  increases  but 
slightly  thereafter.  It  might  be  expected  that  the  efficiency 
would  approach  a  certain  value  as  a  limit  as  the  head  was  indefi- 
nitely increased,  provided  the  bearings  were  adapted  to  the  higher 
speeds.  Such  might  be  the  case  if  it  were  not  for  another  factor. 
The  absolute  velocity  of  discharge,  F2,  varies  as  the  square  root 
of  the  head.  For  low  heads  the  water  discharged  from  the  buck- 
ets strikes  the  case  and  falls  into  the  tail  race  without  interfering 
with  the  wheel.  For  high  heads  it  was  observed  that  the  water 
was  deflected  back  from  the  case  with  sufficient  velocity  to  strike 
the  wheel  and  thus  to  greatly  increase  the  values  of  friction  and 
windage  over  the  values  given  in  Table  12,  where  no  water  was 
present.  The  head  at  which  this  interference  began  to  take  place 
was  lower  as  the  nozzle  opening  was  increased  and  varied  from 
160  ft.  with  the  nozzle  open  three  turns  to  50  ft.  with  the  nozzle 

256 


TEST  DATA  257 

wide  open.  This  would  account  for  the  decrease  in  efficiency 
found  under  the  high  heads  and  would  also  account  to  some 
extent  for  the  maximum  efficiency  being  found  at  different  nozzle 
openings  under  the  various  heads.  These  last  facts  would  have 
no  application  for  the  reaction  turbine. 

Some  test  data  taken  for  the  Pelton  Water  Wheel  Co.  by  the 
J.  G.  White  Co.  will  be  found  in  Table  13.  The  results  of  tests 
on  two  centrifugal  pumps  of  widely  different  types  are  given  in 
Tables  14  and  15. 

In  the  construction  of  characteristic  curves,  the  following 
method  has  been  found  to  be  very  convenient.  Construct  curves 
between  efficiency  and  speed  under  1-ft.  head  for  the  various 
gate  openings.  For  any  given  efficiency  the  speeds  for  the  dif- 
ferent gates  can  be  obtained  from  these  and  the  points  thus 
dermined  location  on  the  characteristic  curve.  The  iso-effi- 
ciency  curves  may  be  drawn  through  these  points,  thus  eliminat- 
ing the  necessity  of  interpolation.  Smooth  efficiency  curves, 
however,  should  be  drawn,  since  very  slight  errors  in  data  appear 
magnified  on  the  characteristic  curve. 


17 


258 


HYDRAULIC  TURBINES 


TABLE  1. — TESTS  OP  A  32-iNCH  R.  H.  WELLMAN-SEAVER-MORGAN 
COMPANY  TURBINE  WHEEL,  No.  1795 

Date,  February  18  and  19,  1909.     Case  No.  1794 
Wheel  supported  by  ball-bearing  step.     Swing-gate.     Conical  draft-tube 


Number  of  experi- 
ment 

Porportional 
part  of 

Head  acting  on  wheel, 
in  feet 

Duration  of  experi- 
ment, in  minutes 

1 

a 
§  % 

o   2 
•43  o 

11 

Quantity  of  water 
discharged  by  wheel, 
in  cubic  feet  per 
second 

Horse-power  devel- 
oped by  wheel 

Percentage  of  effi- 
ciency of  wheel 

Percentage  of 
full  opening  of 
speed-gate 

Percentage  of 
full  discharge 
of  wheel 

•69 

1.000 

1.024 

17.23 

4 

112.50 

34.45 

46.84 

69.59 

67 

1.000 

1.009 

17.50 

3 

152.67 

34.22 

55.80 

82.16 

68 

1.000 

1.008 

17.57 

2 

163.00 

34.05 

57.31 

83.98 

66 

1.000 

1.005 

17.46 

3 

164.00 

34.05 

56.90 

84.40 

65 

'    .000 

0.992 

17.47 

3 

172.67 

33.60 

55.92 

84.00 

64 

.000 

0.979 

17.48 

3 

181.33 

33.17 

54.53 

82.93 

63 

.000 

0.965 

17.46 

4 

189.25 

32.67 

52.53 

81.21 

62 

.000 

0.946 

17.47 

4 

213.00 

32.06 

49.27 

77.57 

61 

.000 

0.688 

17.59 

4 

251.50 

23.39 

34.91 

74.81 

60 

1.000 

0.631 

17.86 

4 

298.00 

21.62 

0.00 

0.00 

59 

0.889 

0.933 

17.50 

3 

113.67 

31.63 

45.23 

72.04 

58 

0.889 

0.934 

17.52 

3 

135.67 

31.70 

50.84 

80.72 

57 

0.889 

0.931 

17.54 

3 

146.33 

31.60 

52.80 

84.01 

56 

0.889 

0.925 

17.53 

3 

153.67 

31.40 

53.32 

85.41 

55 

0.889 

0.922 

17.44 

4 

156.50 

31.21 

52.85 

85.62 

54 

0.889 

0.916 

17.45 

3 

160.33 

31.00 

52.66 

85.84 

53 

0.889 

0.909 

17.46 

3 

164  .  00 

30.79 

52.35 

85.87 

52 

0.889 

0.903 

17.47 

4 

168.00 

30.59 

52.07 

85.92 

51 

0.889 

0.898 

17.48 

4 

172.75 

30.42 

51.95 

86.14 

50 

0.889 

0.893 

17.48 

4 

176.75 

30.25 

51.52 

85.91 

49 

0.889 

0.886 

17.49 

4 

184.00 

30.05 

51.08 

85.69 

48 

0.889 

0.864 

17.53 

4 

206  .  50 

29.31 

47.77 

81.98 

47 

0.889 

0.795 

17.59 

4 

244  .  75 

27.02 

33.97 

63.02 

46 

0.889 

0.572 

17.91 

4 

298.75 

19.61 

0.00 

0.00 

45 

0.741 

0.809 

17.57 

4 

106.00 

27.48 

39.23 

71.65 

44 

0.741 

0.801 

17.57 

4 

139.50 

27.20 

45.18 

83.35 

43 

0.741 

0.791 

17.55 

3 

152.00 

26.85 

45.71 

85.53 

41 

0.741 

0.784 

17.57 

4 

159.75 

26.63 

45.82 

86.35 

40 

0.741 

0.780 

17.60 

4 

166.25 

26.52 

46.15 

87.18 

42 

0.741 

0.777 

17.56 

4 

171.50 

26.40 

•46.02 

87.53 

39 

0.741 

0.770 

17.64 

4 

180.75 

26.22 

45.99 

87.68 

38 

0.741 

0.758 

17.67 

4 

191.50 

25.82 

44.30 

85.61 

37 

0.741 

0.727 

17.73 

3 

210.67 

24.81 

38.99 

78.15 

36 

0.741 

0.693 

17.79 

3 

231.33 

23.68 

32.11 

67.20 

35 

0.741 

0.503 

18.03 

2 

298.00 

17.31 

0.00 

0.00 

34 

0.593 

0.669 

17.84 

3 

101.00 

22.89 

32.71 

70.63 

33 

0.593 

0.660 

17.84 

3 

122.67 

22.61 

36.32 

79.40 

31 

0.593 

0.644 

17.86 

3 

145.00 

22.07 

37.  17 

84.04 

30 

0.593 

0.642 

17.85 

3 

151.33 

21.98 

37.81 

84.97 

32 

0.593 

0.637 

17.88 

4 

158.00 

21.83 

38.081 

85.87 

29 

0.593 

0.632 

17.91 

4 

163.75 

21.68 

37.8 

86.02 

TEST  DATA 


259 


TABLE  1. — (Continued] 


•n 

Proportional 

"if 

•c 

1 

S3  "aS   <3 

*fl> 

, 

1 

part  of 

1 

5  OT 

jq 

11  J 

I 

*o   o 

O    g) 

o 

.a 

•3 

„_,   ^  « 

*!i 

"3  » 

"3 

IH 

II! 

IJI 

|; 

o'2 

Jl 
a 

.2  'g 

IJU 

P.  -0 
1 

i! 

a  u 

i  | 

|il 

I  3* 

§  a 

£  a 

fi 

11*1 

II 

II 

5 

0<  - 

AH  *" 

W  ""* 

w 

A5  ft 

Of'*-  « 

n  ° 

AH    ° 

28 

0.593 

0.622 

17.91 

3 

173.00 

21.33 

36.82 

84.98 

27 

0.593 

0.606 

17.99 

3 

188.00 

20.82 

34.79 

81.90 

26 

0.593 

0.537 

18.16 

4 

239.00 

18.54 

22.11 

57.92 

25 

0.593 

0.406 

18.05 

3 

281  .  00 

13.98 

0.00 

0.00 

24 

0.444 

0.505 

17.91 

3 

94.00 

17.32 

23.92 

67.99 

22 

0.444 

0.489 

17.92 

2 

130.00 

16.78 

27.06 

79.36 

21 

0.444 

0.486 

17.94 

3 

136  .  00 

16.70 

27.68 

81.48 

23 

0.444 

0.486 

17.92 

4 

139  .  25 

16.66 

27.70 

81.82 

20 

0.444 

0.481 

17.94 

3 

145.33 

16.50 

27.57 

82.12 

19 

0.444 

0.472 

17.94 

4 

153.00 

16.21 

26.90 

81.56 

18 

0.444 

0.465 

17.95 

5 

161.40 

15.98 

26.13 

80.34 

17 

0.444 

0.450 

17.99 

4 

178.00 

15.46 

24.71 

78.32 

16 

0.444 

0.402 

18.04 

4 

231.50 

13.85 

16.07 

56.70 

15 

0.444 

0.325 

18.17 

4 

281.25 

11.22 

0.00 

0.00 

14 

0.296 

0.325 

18.19 

3 

89.67 

11.22 

15.35 

66.32 

13 

0.296 

0.320 

18.21 

3 

107.00 

11.06 

16.58 

72.60 

11 

0.296 

0.317 

18.23 

3 

114.67 

10.97 

16.98 

74.85 

12 

0.296 

0.316 

18.22 

3 

118.33 

10.94 

16.97 

75.07 

10 

0.296 

0.313 

18.24 

3 

122  .  67 

10.82 

17.03 

76.07 

9 

0.296 

0.308 

18.25 

4 

130.75 

10.68 

16.94 

76.62 

8 

0.296 

0.304 

18.26 

4 

137.75 

10.54 

16.57 

75.91 

7 

0.296 

0.300 

18.30 

4 

146.50 

10.41 

16.27 

75.29 

6 

0.296 

0.295 

18.31 

4 

154.75 

10.23 

15.75 

74.15 

5 

0.296 

0.291 

18.32 

4 

164.00 

10.10 

15.17 

72.31 

4 

0.296 

0.283 

18.33 

5 

181.20 

9.82 

14.25 

69.81 

3 

0.296 

0.276 

18.36 

4 

191.25 

9.58 

13.27 

66.53 

2 

0.296 

0.266 

18.33 

4 

208  .  00 

9.23 

11.55 

60.18 

1 

0.296 

0.225 

18.37 

4 

261.00 

7.83 

0.00 

0.00 

NOTE. — For  experiments  Nos.  1,  15,  25,  35,  46  and  60,  the  jacket  was  loose. 

During  the  above  experiments,  the  weight  of  the  dynamometer,  and  of  that  portion  of 
the  shaft  which  was  above  the  lowest  coupling  was  1,300  Ib. 

With  the  flume  empty,  a  strain  of  1.0  Ib.,  applied  at  a  distance  of  2.4  ft.  from  the  center 
of  the  shaft,  sufficed  to  start  the  wheel. 


260 


HYDRAULIC  TURBINES 


TABLE  2.— TESTS  OF  A  28-iNCH  R.  H.  WELLMAN-SEAVER-MORGAN 
COMPANY  TURBINE  WHEEL,  No.  1796 

Date,  February  25,  1909 
Wheel  supported  by  ball-bearing  step.     Swing-gate.      Conical  draft-tube 


.i 

Proportional 

1 

.i 

13 

Q)     *^       Q) 

^1, 

H 

1 

part  of 

ft  J 

Jl 

"a  43   a 

> 

"o  *o 

•8  8, 

if 

a 
o 

o  "3 
| 

"S 

*   *1 

73  1 

"o  8 

"o 

H 

e  1 

1 

*o  2 

lis 

S3  -g 

«  ^ 

1- 

I  ll 

|i| 

ll 

•£  -^ 

I** 

'•3  43    o    fl 

S  b 

ft42 

i  -a 

|1 

a  g 

o  —  •    v 

V    £H      ^ 

"3  «2 

2  g 

>   85 

g    g          S 

£    « 

8  g 

3    a 

K    ^3     <M 

f-t    *2    ^_, 

£  a 

3    a 

S   2 

^•^J3O 

0     £ 

fc    ^ 

PH  ""    " 

AH  *~ 

H<a 

Q  e 

«   ft 

a 

w  ° 

£ 

95 

1.077 

0.971 

17.11 

4 

153.00 

97.00 

125.20 

66.52 

94 

1.077 

.017 

16.97 

3 

199  .  67 

101.16 

147.66 

75.81 

93 

1.077 

.036 

16.94 

3 

224  .  33 

102.98 

156.38 

79.04 

92 

1.077 

.053 

16.89 

3 

239  .  33 

104  .  50 

159  .  58 

79.72 

91 

1.077 

.061 

16.87 

3 

247.33 

105.22 

161  .  17 

80.06 

89 

1.077 

.068 

16.81 

3 

253.67 

105.70 

161.45 

80.12 

90 

1.077 

.072 

16.80 

3 

259  .  00 

106.08 

161.71 

80.01 

88 

1.077 

.079 

16.82 

4 

267.50 

106.82 

162.15 

79.58 

87 

1.077 

.026 

17.05 

3 

294.67 

102.27 

125.03 

63.23 

86 

1.000 

0.913 

17.28 

3 

147.00 

91.65 

120.29 

66.98 

85 

1.000 

0.957 

17.16 

2 

190.50 

95.70 

144  .  34 

77.50 

84 

1.000 

0.972 

17.14 

3 

211.67 

97.10 

152.69 

80.89 

83 

1.000 

0.981 

17.13 

3 

225.00 

98.03 

156.85 

82.36 

82 

1.000 

0.990 

17.11 

3 

232.67 

98.90 

157.96 

82.31 

80 

1  .  000 

0.996 

17.09 

4 

240.25 

99.43 

160.19 

83.13 

81 

1.000 

1.003 

17.07 

3 

247  .  33 

100.07 

161.17 

83.19 

79 

1.000 

1  .004 

17.07 

4 

252  .  25 

100.14 

160,55 

82.82 

•     78 

1.000 

1.001 

17.13 

4 

259  .  00 

100.00 

157.00 

80.82 

77 

1.000 

0.983 

17.22 

3 

268.33 

98.43 

146.39 

76.15 

76 

1.000 

0.911 

17.47 

4 

293  .  50 

91.96 

106.75 

58.59 

106 

0.923 

0.868 

17.32 

4 

143.25 

87.24 

115.49 

67.39 

105 

0.923 

0.899 

17.24 

5 

176.00 

90.12 

135.49 

76.90 

104 

0.923 

0.920 

17.15 

4 

201.00 

91.96 

146.21 

81.74 

101 

0.923 

0.931 

16.93 

5 

213.20 

92.52 

148.62 

83.66 

100 

0.923 

0.936 

16.93 

6 

220.67 

92.96 

150.48 

84.31" 

99 

0.923 

0.942 

16.91 

4 

227.25 

93.51 

151.52 

84.50 

102 

0.923 

0.945 

16.93 

4 

232.00 

93.82 

151.88 

84.31 

103 

0.923 

0.945 

17.04 

4 

235.50 

94.14 

152.74 

83.96 

98 

0.923 

0.945 

16.93 

4 

237.75 

93.82 

151.32 

84.00 

97 

0.923 

0.924 

17.02 

4 

254  .  50 

92.04 

138.84 

78.15 

96 

0.923 

0.823 

17.25 

4 

288.25 

82.47 

87.36 

54.15 

42 

0.923 

0.870 

17.17 

3 

146.33 

87.02 

116.20 

68.57 

41 

0.923 

0.895 

17.10 

4 

170.00 

89.37 

130.87 

75.51 

40 

0.923 

0.921 

17.04 

4 

202.75 

91.80 

146.25 

82.44 

39 

0.923 

0.928 

16.99 

3 

208  .  67 

92.35 

147.99 

83.17 

35 

0.923 

0.932 

17.03 

4 

216.25 

92.81 

150.75 

84.10 

38 

0.923 

0.937 

16.97 

4 

220.00 

93.20 

151.36 

84.38 

36 

0.923 

0.939 

17.01 

4 

223.75 

93.44 

152.58 

84.65 

34 

0.923 

0.940 

17.02 

4 

226.25 

93.66 

150.86 

83.45 

37 

0.923 

0.944 

16.97 

3 

238.33 

93.90 

151.69 

83.  9t 

33 

0.923 

0.921 

17.13 

4 

256.25 

92.05 

139.80 

78.18 

32 

0.923 

0.823 

17.27 

3 

288.00 

82.54 

87.29 

53.99 

TEST  DATA 


261 


TABLE  2 . — (Continued) 


Number  of  experi- 
ment 

Proportional 
part  of 

Head  acting  on  wheel, 
in  feet 

Duration  of  experi- 
ment, in  minutes 

Revolutions  of  wheel 
per  minute 

Quantity  of  water 
discharged  by  wheel, 
in  cubic  feet  per 
second 

Horse-power  devel- 
oped by  wheel 

Percentage  of  effi- 
ciency of  wheel 

Percentage  of 
full  opening  of 
speed-gate 

Percentage  of 
full  discharge 
of  wheel 

31 

0.923 

0.730 

17.50 

4 

334  .  75 

73  .  75 

0.00 

0.00 

74 

0.846 

0.824 

17.46 

3 

158.67 

83.15 

120.23 

73.02 

75 

0.846 

0.836 

17.46 

3 

175.67 

84.35 

129.91 

77.78 

72 

0.846 

0.861 

17.34 

5 

202.20 

86.50 

143.40 

84.30 

TO 

0  .  846 

0.865 

17.33 

4 

209.00 

86.95 

145.69 

85.25 

71 

0  .  846 

0.868 

17.33 

3 

215.00 

87.24 

147.27 

85.89 

73 

0.846 

0.870 

17.34 

4 

219.25 

87.47 

148.19 

86.15 

69 

0.846 

0.869 

17.32 

4 

221.25 

87.32 

147.53 

86.01 

68 

0  .  846 

0.866 

17.33 

4 

227.75 

87.02 

144.96 

84.76 

67 

0.846 

0.858 

17.36 

4 

231.75 

86.25 

140.48 

82.73 

66 

0.846 

0.845 

17.39 

4 

243  .  75 

85.11 

132.98 

79.22 

65 

0.846 

0.828 

17.44 

4 

256.50 

83.44 

124.39 

75.37 

64 

0.846 

0.754 

17.59 

3 

282.00 

76.31 

85.47 

56.15 

30 

0.769 

0.750 

17.38 

3 

141.00 

75.52 

102.56 

68.90 

'S6 

0.769 

0.766 

17.35 

3 

166.00 

77.02 

115.72 

76.36 

27 

0.769 

0.779 

17.32 

3 

183.00 

78.24 

124.24 

80.84 

25 

0.769 

0.789 

17.31 

3 

194.00 

79.25 

129.36 

83.15 

29 

0.769 

0.793 

17.25 

4 

200.75 

79.48 

131.42 

84.52 

28 

0.769 

0.792 

17.26 

4 

206.00 

79.40 

131.11 

84.36 

24 

0.769 

0.773 

17.38 

4 

226.25 

77.82 

123.43 

80.47 

23 

0.769 

0.735 

17.49 

3 

251.33 

74.17 

106.64 

72.49 

-22 

0.769 

0.690 

17.56 

3 

269  .  67 

69.82 

81.73 

58,78 

21 

0.769 

0.623 

17.68 

4 

323.75 

63.27 

0.00 

0.00 

20 

0.615 

0.617 

17.91 

4 

139.50 

63.07 

88.79 

69.31 

16 

0.615 

0.627 

17.79 

3 

158.33 

63.81 

95.97 

74.55 

17 

0.615 

0.634 

17.80 

3 

171.33 

64.55 

101.26 

77.71 

15 

0.615 

0.638 

17.73 

4 

179  .  50 

64.82 

103.37 

79.31 

18 

0.615 

0.636 

17.77 

4 

183.00 

64.74 

103.16 

79.07 

19 

0.615 

0.634 

17.80 

2 

188.00 

64.61 

102.56 

78.64 

14             0.615 

0.627 

17.72 

4 

194.50 

63.74 

100.21 

78.24 

13             0.615 

0.596 

17.77 

3 

218.67 

60.60 

92.78 

75.97 

12 

0.615 

0.563 

17.79 

4 

243.00 

57.29 

73.65 

63.72 

11 

0.615 

0.519 

17.92 

4 

312.00 

53.00 

0.00 

0.00 

10 

0.462 

0.452 

17.34 

3 

117.33 

45.40 

56.90 

63.73 

5 

0.462 

0.453 

17.02 

3 

136.00 

45.65 

61.83 

70.17 

7 

0  .  462 

0.461 

17.08 

4 

146.75 

46.04 

64.94 

72.82 

6 

0.462 

0.462 

17.05 

4 

152.00 

46.04 

66.34 

74.52 

4 

0.462 

0.462 

16.92 

4 

155.25 

45.90 

65.88 

74.79 

9 

0.462 

0.459 

17.16 

3 

162.00 

45.94 

66.78 

74.69 

8 

0.462 

0.457 

17.15 

3 

166.67 

45.69 

65.67 

73.90 

3 

0.462 

0.451 

16.98 

4 

172.25 

44.83 

62  .  65 

72.57 

2 

0.462 

0.432 

17.05 

4 

217.50 

43.04 

52.74 

63.37 

1 

0.462 

0.404 

17.18 

5 

282.40 

40.40 

0.00 

0.00 

NOTE. — During  the  above  experiments,  the  weight  of  the  dynamometer,  and  of  that 
portion  of  the  shaft  which  was  above  the  lowest  coupling,  was  2,600  Ib. 

With  the  flume  empty,  a  strain  of  0. 5  Ib.,  applied  at  a  distance  of  3. 2  ft.  from  the  center 
at  the  shaft,  sufficed  to  start  the  wheel. 


262 


HYDRAULIC  TURBINES 


TABLE  3. — TESTS  OF  A  30-iNCH  R.  H.  WELLMAN-SE AVER-MORGAN 
COMPANY  TURBINE  WHEEL,  No.  1797 

Date,  February  26  and  27,  1909 
Wheel  supported  by  ball-bearing  step.     Swing-gate.     Conical  draft-tube 


1 

Proportional 

1 

.£ 

J" 

fr-S| 

13 

• 

1 

part  of 

Ja 

P 

a  S 

*  J3    ft 

o> 

M 

S 

«*"      •*-!                                         «*«        0> 

3.          I 

a 
o 

*! 

•3 

..'I! 

1l 

«*.    « 

M 

T3 

t* 

0 

I'g  8        I  2  •« 

a 

a 

g* 

§  | 

*H 

I   >> 

o 

11 

ill    SU"8 

!* 

ii 

^         fH 

II 

fill 

fl     W           O 

3-2     0     S 

li 

1  1 

3    S 

fir.-*      1  ^  ° 

W  .2 

P     * 

rt  ^ 

§*.a  S 

w  ° 

£'5 

82 

.000 

0.942 

17.06 

3 

149.00 

94.89 

126.41 

68.85 

81 

.000 

0.950 

16.99 

3 

160.33 

95.52 

131.16 

71.26 

80 

.000 

0.963 

17.00 

4 

190.50 

96.85 

144.30 

77.28 

75 

.000 

0.977 

17.02 

4 

210.50 

98.34 

150.52 

79.30 

74 

.000 

0.984 

16.95 

4 

220.00 

98.81 

153.31 

80.72 

76 

.000 

0.992 

17.01 

4 

227  .  25 

99.75 

155.61 

80.87 

78 

.000 

0.994 

16.94 

4 

229  .  50 

99.75 

155.76 

81.28 

77 

.000 

0.997 

16.98 

4 

231.25 

100.14 

155.55 

80.66 

79 

.000 

1.002 

16.90 

3 

239.67 

100.45 

156.85 

81.47 

73 

1.000 

1.003 

16.94 

4 

254.75 

100.70 

154.37 

79.80 

72 

1.000 

0.871 

17.13 

3 

287.00 

87.93 

86.96 

50.91 

71 

.  1.000 

0.728 

17.44 

3 

317.67 

74.17 

0.00 

0.00 

52 

0.923 

0.875 

17.08 

3 

131.33 

88.14 

111.42 

65.26 

51 

0.923 

0.896 

16.95 

3 

156.33 

89.97 

127.89 

73.95 

50 

0.923 

0.913 

16.93 

4 

188.25 

91.66 

142  .  59 

81.02 

44 

0.923 

0.920 

16.89 

4 

201.25 

92.20 

146  .  34 

82.86 

45 

0.923 

0.927 

16.85 

4 

211.50 

92.81 

148.67 

83.83 

46 

0.923 

0.931 

16.86 

4 

217.00 

93.20 

149.91 

84.12 

49 

0.923 

0.934 

16.83 

4 

220.50 

93.42 

150.99 

84.68 

47 

0.923 

0.936 

16.83 

4 

223.75 

93.60 

151.86 

85.00 

48 

0.923 

0.937 

16.83 

3 

226.33 

93.74 

150.87 

84.32 

43 

0.923 

0.929 

16.88 

4 

241.75 

93.13 

146.49 

82.17 

42 

0.923 

0.788 

17.21 

4 

276.25 

79.76 

83.70 

53.77 

41 

0.923 

0.670 

17.44 

4 

312.50 

68.26 

0.00 

0.00 

70 

0.846 

0.868 

17.07 

4 

192.75 

87.47 

142.50 

84.15 

67 

0.846 

0.875 

17.10 

4 

205.00 

88.22 

147.83 

86.41 

68 

0.846 

0.876 

17.06 

4 

209  .  50 

88.22 

148.53 

87.02 

66 

0.846 

0.876 

17.11 

4 

212.00 

88.37 

149  .  02 

86.91 

69 

0.846 

0.877 

17.05 

3 

213.00 

88.30 

148.43 

86.94 

65 

0.846 

0.876 

17.13 

5 

213.80 

88.37 

147.70 

86.03 

64 

0.846 

0.875 

17.13 

5 

215.80 

88.30 

146.46 

85.38 

63 

0.808 

0.810 

17.33 

4 

143.50 

82.24 

114.78 

71.01 

62 

0.808 

0.834 

17.25 

3 

177.00 

84.44 

135.14 

81.81 

61 

0.808 

0.843 

17.24 

5 

194  .  80 

85.35 

142.83 

85.59 

60 

0.808 

0.847 

17.21 

3 

201.00 

85.73 

144.94 

86.62 

59 

0.808 

0.848 

17.19 

3 

206.00 

85.73 

146.05 

87.39 

58 

0.808 

0.847 

17.19 

4 

208.25 

85.65 

145.12 

86.91 

57 

0.808 

0.845 

17.19 

3 

210.00 

85.41 

143.80 

86.36 

56 

0.808 

0.840 

17.22 

3 

216.67 

85.05 

141.80 

85.37 

55 

0.808 

0.825 

17.25 

4 

224.75 

83.53 

136.19 

83.34 

54 

0.808 

0.807 

17.29 

4 

236.50 

81.80 

128.98 

80.41 

TEST  DATA 


263 


TABLE  3.— (Continued} 


«ll 

Proportional 

"3 

•A 

_^ 

M  _r  fc( 

J< 

jrj 

1 

part  of 

8 
jfl 

|S 

S 

J3 

5  j  s 

• 

E 

• 
O 

H 

•3  "3 

M 

o  a 

p 
o 

8    3 

a 

* 

"8 

*  *1 

| 

•si 

"3 

L 

.8     - 

f.s  <§ 
tj 

rfi 

ti 

*3  £ 

§    3 
'•3  .S 

JS     g 

Ili, 

A  -^ 

i: 

6  g 

§  £2    $ 

s  ^  ^ 

"S  "" 

g  1    ' 

5  *« 

g  g     I 

2  S 

g  d 

3   a 

fe  *2    ^* 

fe    *"3    **•* 

g   d 

3   a 

^  ^ 

o  ft 

t-    fQ> 

55   B 

&<  •** 

OH  "*" 

W  "* 

Q  * 

PH    ft 

^TJ  .„     0) 

a  ° 

&  ° 

53 

0.808 

0.695 

17.51 

4 

264  .  25 

70.97 

80.06 

56.81 

40 

0.769 

0.755 

17.29 

4 

118.50 

76.53 

93.35 

62.21 

39 

0.769 

0.776 

17.21 

4 

146.50 

78.47 

110.97 

72.46 

35 

0.769 

0.800 

17.03 

4 

174.75 

80.49 

127.07 

81.74 

36 

0.769 

0.806 

17.03 

4 

190.75 

81.13 

135.24 

86.31 

37 

0.769 

0.805 

17.10 

4 

197.25 

81.21 

137.46 

87.28 

38 

0.769 

0.805 

17.15 

4 

200.50 

81.28 

137.29 

86.84 

34 

0.769 

0.803 

16.83 

4 

199.50 

80.34 

132.98 

86.72 

33 

0.769 

0.788 

16.93 

4 

211.75 

79.04 

128.32 

84.55 

32 

0.769 

0.745 

16.93 

4 

232.50 

74.72 

112.71 

78.56 

31 

0.769 

0.660 

17.36 

4 

257.50 

67.02 

78.02 

59.13 

30 

0.769 

0.579 

17.60 

3. 

306.00 

.  59.27 

0.00 

0.00 

29 

0.615 

0.610 

17.69 

3 

100.67 

62.60 

69.54 

55.37 

28 

0.615 

0.621 

17.64 

3 

127.67 

63.60 

85.10 

66.89 

25 

0.615 

0.643 

17.60 

3 

167.33 

65.76 

104.44 

79.57 

27 

0.615 

0.644 

17.57 

4 

175.50 

65.83 

107.41 

81.89 

24 

0.615 

0.643 

17.61 

4 

178.00 

65.76 

107.86 

82.13 

26 

0.615 

0.641 

17.59 

3 

179.67 

65.55 

106.70 

81.60 

23 

0.615 

0.482 

17.96 

3 

296.67 

49.84 

0.00 

0.00 

20 

0.615 

0.634 

17.88 

4 

155.00 

65.37 

101.44 

76.53 

21 

0.615 

0.634 

17.80 

5 

151.40 

65.22 

98.17 

74.56 

22 

0.615 

0.639 

17.69 

3 

160.33 

65.50 

102.98 

78.37 

19 

0.615 

0.644 

18.08 

5 

174.20 

66.80 

111.89 

81.69 

18 

0.615 

0.643 

18.11 

3 

180.00 

66.73 

112.35 

81.97 

17 

0.615 

0.638 

18.07 

4 

183.50 

66.19 

111.20 

81.98 

16 

0.615 

0.636 

18.03 

4 

185.00 

65.90 

109  .  86 

81.53 

15 

0.615 

0.630 

17.96 

3 

190.00 

65.15 

109  .  38 

82.43 

14 

0.615 

0.609 

17.81 

4 

201.00 

62.66 

103.53 

81.80 

13 

0.615 

0.574 

17.79 

4 

216.75 

59.07 

91.94 

77.15 

12 

0.615 

0.553 

17.54 

5 

245.20 

56.46 

74.29 

66.15 

11 

0.615 

0.483 

17.71 

5 

294.60 

49.58 

0.00 

0.00 

8 

0.462 

0.475 

17.82 

3 

104.33 

48.95 

58.16 

58.80 

7 

0.462 

0.483 

17.75 

4 

139.75 

49.58 

71.98 

72.12 

6 

0.462 

0.482 

17.76 

3 

147.00 

49.51 

73.93 

74.14 

5 

0.462 

0.481 

17.80 

4 

156.75 

49.45 

75.99 

76.12 

9 

0.462 

0.477 

17.93 

2 

162.00 

49.22 

76.57 

76.51 

10 

0.462 

0.472 

17.97 

3 

167.00 

48.78 

75.90 

76.35 

4 

0.462 

0.464 

17.88 

5 

173.60 

47.80 

73.64 

75.97 

3 

0.462 

0.436 

18.05 

4 

216.50 

45.13 

65.60 

71.01 

2 

0.462 

0.415 

18.08 

4 

246  .  25 

43.02 

44.77 

50.75 

1 

0.462 

0.381 

18.27 

4 

280.50 

39.75 

0.00 

0.00 

NOTE.— The  jacket  was  loose  for  Experiments  Nos.  1,  11,  23,  30,  41,  and  71. 

During  the  above  experiments,  the  weight  of  the  dynamometer,  and  of  that  portion  of 
the  shaft  which  was  above  the  lowest  coupling  was  2,600  Ib. 

With  the  flume  empty,  a  strain  of  0.5  Ib.,  applied  at  a  distance  of  3.2  ft.  from  the  center 
of  the  shaft,  sufficed  to  start  the  wheel. 


264 


HYDRAULIC  TURBINES 


TABLE  4. — TESTS  OF  A  31-iNCH  R.  H.  WELLMAN-SEAVER-MORGAN 
COMPANY  TURBINE  WHEEL,  No.  1799 

Date,  March  2  and  3,  1909 
Wheel  supported  by  ball-bearing  steps.     Swing-gate.      Conical  draft-tube 


•g 

Proportional 

IB 

.i 

1 

s  ^  s 

^ 

£ 

part  of 

1 

.  1$ 

J 

^. 

11 

r> 

OJ 

p 

| 

1"8 

..•SB 

£ 
a 

O 

f| 

!> 
*0 

£* 

^   %£ 

73  —  . 

£ 

«*-  *^ 
°.S 

Number  of 
ment 

Percentage 
full  opening 
speed-gate 

Percentage 
full  discha 
of  wheel 

Head  acting 
in  feet 

Duration  ol 
ment,  in  ir 

Revolutions 
per  minute 

Quantity  o 
discharged  1 
in  cubic  : 
second 

Horse-power 
oped  by.  wh 

Percentage 
ciency  of  w 

49 

.000 

1.049 

17.15 

3 

134.67 

79.75 

115.81 

74.66 

48 

.000 

1.038 

17.15 

3 

147.00 

78.95 

119.29 

77.68 

47 

.000 

1.025 

17.19 

3 

165.67 

78.02 

123.40 

81.13 

46 

.000 

1.024 

17.18 

2 

174.00 

77.91 

124  .  34 

81.91 

45 

.000 

1.018 

17.19 

3 

178.33 

77  .  52 

124.19 

82.18 

44 

1.000 

1.013 

17.19 

3 

183.33 

77.16 

124.35 

82.66 

43 

1.000 

0.012 

17.19 

4 

186.25 

77.09 

124.07 

82.56 

42 

1.000 

1.009 

17.18 

4 

189  .  50 

76.82 

123.94 

82.81 

41 

.000 

1.007 

17.19 

'  3 

193.00 

76.67 

123.89 

82.89 

40 

.000 

1.006 

17.12 

4 

195.50 

76.44 

123.13 

82.96 

39 

.000 

1.002 

17.09 

4 

200.25 

76.10 

122.48 

83.04 

38 

.000 

0.999 

17.10 

4 

206  .  25 

75.87 

122.41 

83.19 

37 

.000 

0.997 

17.07 

3 

210.33 

75.66 

121.01 

82.62 

36 

.000 

0.993 

17.09 

4 

219.00 

75.38 

119.36 

81.70 

35 

.000 

0.906 

17.24 

4 

258.75 

'      69.11 

78.35 

57.98 

34 

.000 

0.744 

17.45 

4 

302.25 

57.07 

0.00 

0.00 

32 

0.883 

0.909 

17.18 

3 

129.33 

69.23 

101  .  03 

74.90 

31 

0.883 

0.904 

17.20 

4 

154.25 

68.82 

111.16 

82.81 

33 

0.883 

0.901 

17.19 

4 

165.00 

68.60 

113.91 

85.18 

30 

0.883 

0.896 

17.20 

3 

172.33 

68.25 

114.80 

86.23 

29 

0.883 

0.892 

17.21 

4 

181.50 

67.97 

116.51 

87.82 

27 

0.883 

0.888 

17.25 

3 

188.00 

67.77 

117.27 

88.45 

26 

0.883 

0.886 

17.30 

4 

194.00 

67.71 

117.49 

88.44 

28 

0.883 

0.885 

17.23 

4 

197.25 

67.43 

117.06 

88.85 

25 

0.883 

0.883 

17.31 

4 

201  .  75 

67.49 

116.07 

87.61 

24 

0.883 

0.857 

17.28 

4 

213.25 

65.42 

109  .  77 

85.62 

23 

0.883 

0.818 

17.31 

4 

227.00 

62.53 

96.23 

78.39 

22 

0.883 

0.761 

17.42 

4 

244.25 

58.32 

73.96 

64.19 

21 

0.883 

0.620 

17.65 

4 

291.75 

47.83 

0.00 

0.00 

69 

0.750 

0.835 

17.28 

4 

124.75 

63.73 

91.41 

73.19 

68 

0.750 

0.839 

17.23 

4 

148.75 

63.93 

103  .  59 

82.93 

67 

0.750 

0.834 

17.23 

4 

168.75 

63.60 

109.35 

87.99 

64 

'    0.750 

0.830 

17.24 

4 

179  .  25 

63.26 

110.72 

89.52 

65 

0.750 

0.829 

17.23 

4 

182.00 

63.20 

110.77 

89.69 

63 

0.750 

0.828 

17.25 

4 

186.25 

63.12 

111.66 

90.43 

66 

0.750 

0.824 

17.23 

3 

188.67 

62.85 

110.83 

90.24 

62 

0.750 

0.821 

17.29 

4 

191.75 

62.72 

110.32 

89.70 

61 

0.750 

0.809 

17.30 

4 

197.50 

61.81 

107.64 

88.76 

60 

0.750 

0.796 

17.34 

4 

203  .  25 

60.86 

104.62 

87.42 

59 

0.750 

0.768 

17.38 

4 

212.50 

58.80 

96.52 

83.28 

58 

0.750 

0.693 

17.55 

4 

237.00 

53.31 

71.76 

67.64 

57 

0  .  750 

0.573 

17.75 

4 

287.00 

44.32 

0.00 

O.'OO 

TEST  DATA 


265 


TABLE  4. — (Continued) 


,i 

Proportional 

1 

.A 

13 

g-s  s 

"3 

Efi 

1 

part  of 

J 

a  S 

1 

1  ^  a 

1 

"8  "8 

•8  S, 

§ 

Q>  2 
g 

"o 

H] 

^  1 

-3  | 

"8 

H 

o>  .9  5 

tl 

•  -Si 

H 

a 
$ 

"8  '§ 

a  % 
.1   S 

S|i 

if 

|? 

1  ^ 

fl    o  *^ 

S^.3 

II 

•_2  _p- 

"S  '3 

'£  j~    o    £3 

ft  X! 

a  -^1 

1  S 

8  ~  * 

2  § 

"o   . 
>    a3 

|I 

§  g 

Is 

|I  8 

|-° 

*    £3 

B  "* 

Q  S 

^& 

|lil 

£=      I'* 

20 

0.667 

0.726 

17.27 

4 

106.25 

55.43 

70.78 

65.20 

19 

0.667 

0.737 

17.26 

3 

148.67 

56.25 

90.93 

82.59 

18 

0.667 

0.739 

17.25 

4 

161.00 

56.39 

95.55 

86.62 

17 

0.667 

0.738 

17.24 

4 

168.50 

56.31 

96.94 

88.05 

16 

0.667 

0.733 

17.27 

3 

173.67 

55.92 

96.76 

88.35 

15 

0.667 

0  .  722 

17.30 

4 

179  .  25 

55.17 

95.52 

88.25 

14 

0.667 

0.699 

17.34 

4 

189.50 

53.48 

91.81 

87  ,30 

13 

0.667 

0.671 

17.39 

4 

201  .  75 

51.37 

85.52 

84  .42 

11 

0.667 

0.508 

17.63 

4 

280.75 

39.17 

0.00 

0.00 

8 

0.500 

0.554 

17.82 

3 

117.00 

42.95 

60.23 

69.39 

7 

0.500 

0.546 

17.91 

3 

135.00 

42.46 

65.40 

75.84 

10 

0.500 

0.548 

17.67 

3 

151.00 

42.28 

68.58 

80.95 

6 

0.500 

0.548 

18.05 

4 

157.50 

42.72 

71.54 

81.80 

9 

0.500 

0.547 

17.71 

4 

157.00 

42.28 

69.41 

81.73 

5 

0.500 

0.539 

18.12 

5 

167.60 

42.11 

71.05 

82,10 

4 

0.500 

0.512 

18.13 

4 

187.00 

40.02 

67.95 

82.58 

3 

0.500 

0.488 

.18.18 

4 

213.00 

38.21 

58.05 

73.68 

2 

0.500 

0.460 

18.07 

4 

232  .  00 

35.92 

42.15 

57.26 

1 

0.500 

0.402 

18.20 

3 

275.00 

31.50 

0.00 

0.00 

52 

0.333 

0.362 

18.18 

3 

96.00 

28.35 

34.88 

59.68 

55 

0.333 

0.361 

18.01 

3 

177.33 

28.10 

39.08 

68.09 

51 

0.333 

0.348 

18.19 

4 

133.00 

27.25 

40.27 

71.64 

54 

0.333 

0.347 

18.21 

3 

139.67 

27.20 

40.60 

72.28 

53 

0.333 

0.340 

18.22 

3 

143.67 

26.62 

39.15 

71.18 

56 

0.333 

0.333 

18.06 

4 

148.00 

26.00 

37.64 

70.69 

50 

0.3S3 

0.316 

18.26 

3 

201.67 

24.82 

36.64 

71.28 

NOTE. — For  Experiments  Nos.  1,  11,  21,  and  57,  the  jacket  was  loose. 

During  the  above  experiments,  the  weight  of  the  dynamometer  and  of  that  portion  of 
the  shaft  which  was  above  the  lowest  coupling  was  2,600  Ib. 

With  the  flume  empty,  a  strain  of  1.0  Ib.,  applied  at  a  distance  of  3.2  ft.  from  the  center  of 
the  shaft,  sufficed  to  start  the  wheel. 


266 


HYDRAULIC  TURBINES 


TABLE  5. — TESTS  OF  A  31-iNCH  R.  H.  WELLMAN-SE AVER-MORGAN 
COMPANY  TURBINE  WHEEL,  No.  1800 

Date,  March  4  and  5,  1909 
Wheel  supported  on  ball-bearing  step.     Swing-gate.      Conical  draft-tube 


E 

"3 

y 

Proportional 
part  of 

Head  acting  on  wheel, 
in  feet 

Duraton  of  experi- 
ment, in  minutes 

Revolutions  of  wheel 
per  minute 

fill 

S  g      3 
3*  .2  8 

Horse-power  devel- 
oped by  wheel 

Percentage  of  effi- 
ciency of  wheel 

Percentage  of 
full  opening  of 
speed-gate 

Percentage  of 
full  discharge 
of  wheel 

65 

1.000 

1.052 

17.38 

3 

112.33 

64.87 

82.99 

64.91 

64 

1.000 

1.040 

17.41 

3 

133.33 

64.20 

91.24 

71.98 

63 

1.000 

1.025 

17.43 

4 

154.25 

63.33 

96.21 

76.85 

62 

.  000 

1.014 

17.42 

3 

168.67 

62.60 

98.64 

79.76 

61 

.000 

.009 

17.40 

3 

183.33 

62.25 

99.92 

81.34 

60 

.000 

.006 

17.42 

4 

193.25 

62.13 

100.65 

82.00 

59 

.000 

.004 

17.43 

4 

201.00 

62.01 

101  .  03 

82.42 

58 

.000 

.002 

17.44 

4 

207.00 

61.88 

101  .  54 

82.96 

56 

.000 

.000 

17.35 

4 

211.00 

61.65 

100  .  95 

83.22 

57 

.000 

0.999 

17.39 

4 

216.75 

61.65 

101.07 

83.13 

55 

.000 

0.998 

17.36 

4 

221.50 

61.53 

100.60 

83.05 

54 

.000  . 

0.995 

17.38 

4 

247.50 

61.40 

97.42 

80.50 

53 

.000 

0.951 

17.45 

4 

266.75 

58.80 

80.77 

69.41 

52 

.000 

0.730 

17.71 

4 

321.25 

45.44 

0.00 

0.00 

51 

0.883 

0.889 

17.48 

4 

132.25 

55.02 

79.29 

72.69 

50 

0.883 

0.888 

17.48 

4 

147.00 

54.91 

83.68 

76.88 

49 

0.883 

0.885 

17.47 

4 

165.00 

54.71 

88.93 

82.04 

48 

0.883 

0.879 

17.39 

4 

179.25 

54.24 

91.18 

85.24 

47 

0.883 

0.876 

17.42 

5 

189.20 

54.11 

91.66 

85.75 

44 

0.883 

0.873 

17.54 

4 

197.50 

54.11 

93.29 

86.67 

43 

0.883 

0.873 

17.40 

4 

204.25 

53.85 

92.77 

87.30 

45 

0.883 

0.872 

17.40 

3 

209.00 

53.79 

92.40 

87.05 

46 

0.883 

0.871 

17.42 

4 

215.50 

53.79 

92.66 

87.19 

42 

0.883 

0.858 

17.40 

3 

225.00 

52.92 

88.57 

84.81 

41 

0.883 

0.810 

17.45 

3 

248.00 

50.06 

75.09 

75.80 

40 

0.883 

0.609 

17.68 

3 

314.00 

37.87 

0.00 

0.00 

75 

0.733 

0.801 

17.76 

4 

158.50 

49.93 

81.59 

81.13 

73 

0.733 

0.798 

17.73 

3 

183.33 

49.70 

86.60 

86.66 

72 

0.733 

0.797 

17.71 

3 

190.33 

49.63 

87.60 

87.88 

71 

0.733 

0.796 

17.72 

3 

197.00 

49.57 

88.28 

88.62 

74 

0.733 

0.795 

17.73 

3 

199.00 

49.50 

87.97 

88.39 

70 

0.733 

0.793 

17.72 

3 

201.67 

49.37 

87.93 

88.63 

69 

0.733 

0.788 

17.70 

4 

204.50 

49.02 

86.69 

88.10 

68 

0.733 

0.775 

17.71 

4 

213.50 

48.27 

84.04 

86.69 

67 

0.733    • 

0.722 

17.79 

3 

234  .  00 

45.04 

70.85 

77.97 

66 

0.733 

0.544 

18.03 

3 

308.67 

34.15 

0.00 

0.00 

38 

0.667 

0.751 

17.58 

3 

171.00 

46.58 

78.70 

84.75 

37 

0.667 

0.750 

17.57 

4 

184.00 

46.52 

81.34 

87.75 

39 

0.667 

0.749 

17.56 

4 

187.00 

46.45 

81.54 

88.14 

35 

0.667 

0.749 

17.59 

4 

185.50 

46.45 

80.88 

87.29 

36 

0.667 

0.747 

17.59 

4 

188.75 

46.38 

81.73 

88.33 

TEST  DATA 


267 


TABLE  5. — (Continued) 


"T 

Proportional 

J"" 

'E 

- 

«  "o3    a> 

j. 

. 

1, 

part  of 

1  8 

J 

i-2  a 

^3 

V 

"8 

°    i  o> 

U| 

P 
a 

0 

X    g 

"S'l 

•8 
§  5 

•si! 

*! 

.* 

M  '3    t« 

M    g   — 

'Js 

a  5 

•I  a 

>>  ^  -2 

0     >> 

2  "3 

!«> 

f  II 

I    1 

|2 

0   "* 

If 

.tn      03    "g    "0 
"fl    *«      "      O 

11 

1  1 

1  * 

Js  & 

|1° 

g  a 

I] 

1  & 

|«lj|      1 

1° 

£•3 

34 

0.667 

0.742 

17.58 

4 

192.50 

46.02 

80.44 

87.67 

33 

0.667 

0.731 

17.59 

4 

198.25 

45.37 

78.04 

86.22 

32 

0.667 

0.753 

17.70 

3 

127.67 

46.87 

68.04 

72.32 

31 

0.667 

0.752 

17.68 

3 

144.67 

46.76 

72.72 

77.56 

29 

0.667 

0.752 

17.66 

4 

164.25 

46.76 

77.59 

82.85 

30 

0.667 

0.753 

17.68 

3 

178.67 

46.82 

81.15 

86.44 

28 

0.667 

0.750 

17.62 

3 

189.00 

46.57 

82.41 

88.56 

27 

0.667 

0.741 

17.62 

3 

193.67 

46.02 

80.93 

88.00 

26 

0.667 

0.731 

17.64 

3 

201.00 

45.44 

79.12 

87.04 

25 

0.667 

0.725 

17.63 

3 

204.00 

45.05 

77.83 

86.41 

24 

0.667 

0.709 

17.64 

3 

213.33 

44.05 

74.93 

85.03 

23 

0.667 

0.678 

17.70 

4 

224  .  50 

42.23 

67.98 

80.19 

22 

0.667 

0.506 

17.85 

3 

303.33 

31.61 

0.00 

0.00 

21 

0.500 

0.566 

17.72 

2 

124.00 

35.27 

50.31 

70.98 

20 

0.500 

0.560 

17.74 

3 

136.33 

34.87 

52.84 

75.32 

19 

0.500 

0.554 

17.71 

4 

146.50 

34.50 

53.23 

76.82 

18 

0.500 

0.543 

17.72 

4 

155.75 

33.82 

52.82 

77.72 

17 

0.500 

0.535 

17.74 

3 

166.00 

33.31 

52.27 

78.00 

16 

0.500 

0.521 

17.77 

3 

176.67 

32.47 

51.36 

78.48 

15 

0.500 

0.507 

17.79 

3 

189.00 

31.61 

50.36 

78.97 

14 

0.500 

0.494 

17.83 

4 

200.25 

30.87 

48.51 

77.71 

13 

0.500 

0.481 

17.86 

3 

214.00 

30.08 

45.36 

74.45 

12 

0.500 

0.464 

17.89 

4 

228.75 

29.02 

41.56 

70.58 

11 

0.500 

0.380 

18.00 

4 

299  .  50 

23.87 

0.00 

0.00 

8 

0.333 

0.360 

18.09 

4 

112.00 

22.65 

30.52 

65.68 

7 

0.333 

0.351 

18.12 

3 

127.00 

22.10 

31.53 

69.43 

6 

0.333 

0.350 

18.13 

3 

136.00 

22.02 

32.12 

70.95 

5 

0.333 

0.346 

18.22 

4 

141.50 

21.84 

31.71 

70.26 

4 

0.333 

0.340 

18.24 

4 

148.00 

21.47 

31.37 

70.63 

3 

0.333 

0.332 

18.31 

4 

154.75 

21.03 

30.93 

70.82 

2 

0.333 

0.323 

18.25 

4 

165.25 

20.42 

30.02 

71.04 

10 

0.333 

0.307 

18.17 

4 

212.50 

19.33 

28.31 

71.08 

9 

0.333 

0.300 

18.15 

3 

231.33 

18.93 

25.22 

64.72 

1 

0.333 

0.248 

18.35 

4 

287.25 

15.74 

0.00 

0.00 

NOTE. — The  jacket  was  loose  for  Experiments  Nos.  1,  11,  22,  40,  52,  and  66. 

During  the  above  experiments,  the  weight  of  the  dynamometer  and  of  that  portion  of  the 
shaft  which  was  above  the  lowest  coupling  was  2,600  Ib. 

With  the  flume  empty,  a  strain  or  0.5  Ib.,  applied  at  a  distance  of  3.2  ft.  from  the  center 
of  the  shaft,  sufficed  to  start  the  wheel. 


268 


HYDRAULIC  TURBINES 


TABLE  6. — TEST  OF  A  27-iNCH  I.  P.  MORRIS  Co.  REACTION  TURBINE 
AT  CORNELL  UNIVERSITY l 


By  R.  L.  Daugherty,  Feb.,  1914 


Proportional 
part  of 
gate  opening 

Road, 
ft. 

Discharge 
cu.  ft.  per 
sec. 

R.p.m. 

Torque 
ft.  Ib. 

B.h.p, 

Efficiency, 
per  cent. 

0.067 

146.1 

7.4 

600 

143 

16.4 

13.3 

0.067 

146.2 

6.9 

647 

0 

0 

00.0 

0.248 

144.9 

18.7 

0 

2760 

0 

00.0 

0.248 

145.4 

16.4 

600 

1365 

156 

57.6 

0.248 

145.7 

12.5 

845 

0 

0 

00.0 

0.476 

143.9 

27.5 

0 

5390 

0 

00.0 

0.476 

144.3 

25.4 

600 

2760 

318 

76.3 

0.476 

145.2 

18.8 

975 

0 

0 

00.0 

0.600 

142.8 

34.8 

0 

6550 

0 

00.0 

0.600 

143  .  1 

31.8 

600 

3820 

437 

84.5 

0.600 

144.7 

23.0 

1022 

0 

0 

00.0 

0.772 

141.8 

40.2 

0 

7520 

0 

00.0 

0.772 

141.8 

38.8 

600 

4820 

550 

88.0 

0.772 

144.0 

26.8 

1038 

0 

0 

00.0 

1.000 

140.6 

46.3 

0 

8130 

0 

00.0 

1.000 

140.5 

44.5 

600 

5400 

617 

87.0 

1.000 

143.4 

32.4 

1060 

0 

0 

00.0 

xFor  an  account  of  this  test  see  "  Investigation  of  the  Performance  of  a 
Reaction  Turbine,"  Trans.  A.  S.  C.  K,  Vol.  LXXVIII,  p.  1270  (1915). 


TEST  DATA 


269 


TABLE  7. — TEST  OF  A   12-lNCH  PELTON-DOBLE  TANGENTIAL  WATER 

WHEEL  UNDER  A  CONSTANT  PRESSURE  HEAD  OF  8.93  FT.  LENGTH  OF 

BRAKE-ARM  =  14-iN. 


Turns  of 
needle 

Head, 
ft. 

Discharge, 
cu.  ft.  per 
sec. 

R.p.m. 

Brake 
load, 
lb. 

B.h.p. 

Efficiency, 
per  cent. 

1 

8.932 

0.0175 

0 

0.48 

0.00000 

00.0 

100 

0.32 

0.00712 

40.0 

150 

0.24 

0.00802 

45.0 

200 

0.16 

0.00712 

40.0 

250 

0.06 

0.00334 

18.8 

280 

0.00 

0.00000 

00.0 

2 

8.936 

0  .  0300 

0 

0.95 

0.0000 

00.0 

100 

0.71 

0.0158 

51.8 

150 

0.58 

0.0194 

63.7 

200 

0.43 

0.0191 

62.8 

250 

0.27 

0.0150 

49.2 

320 

0.00 

0.0000 

00.0 

3 

8.943 

0.0448 

0 

1.50 

0.0000 

00.0 

100 

1.11 

0.0247 

54.2 

150 

0.90 

0.0301 

66.0 

200 

0.67 

0.0298 

65.3 

250 

0.42 

0  .  0234 

51.3 

330 

0.00 

0.0000 

00.0 

4 

8.951 

0.0575 

0 

1.80 

0.0000 

00.0 

100 

1.38 

0.0347 

59.3 

150 

1.16 

0  .  0388 

66.3 

200 

0.90 

0.0400 

68.3 

250 

0.60 

0.0334 

57.0 

340 

0.00 

0.0000 

00.0 

5 

8.959 

0  .  0670 

0 

2.00 

0.0000 

00.0 

100 

1.56 

0.0347 

50.8 

150 

1.32 

0.0441 

64.6 

200 

1.06 

0.0472 

69.2 

250 

0.82 

0.0457 

67.0 

360 

0.00 

0.0000 

00.0 

6 

8.966 

0.0750 

0 

2.14 

0.0000 

00.0 

100 

1.69 

0.0376 

49.2 

150 

1.46 

0.0488 

63.8 

200 

1.19 

0.0526 

68.8 

•  |  ;  .« 

250 

0.90 

0.0502 

65.7 

360 

0.00 

0.0000 

00.0 

7.85 

8.978 

0.0860 

0 

2.60 

0.0000 

00.0 

100 

1.89 

0.0422 

48.0 

150 

1.66 

0.0555 

63.2 

200 

1.34 

0.0596 

67.8 

250 

1.03 

0.0574 

65.0 

360 

0.00 

0.0000 

00.0 

270 


HYDRAULIC  TURBINES 


TABLE  8. — TEST  OF  A  12-iNCH  PELTON-DOBLE  TANGENTIAL  WATER 
WHEEL  UNDER  A  CONSTANT  PRESSURE  HEAD  OF  62.5  FT. 


Turns 
<-  of 
needle 

Head, 
ft. 

Discharge, 
cu.  ft.  per 
sec. 

R.p.m. 

Brake 
load, 
Ib. 

B.h.p. 

Efficiency, 
per  cent. 

1 

62.52 

0.041 

100 

3.10 

0.069 

22.8 

200 

2.85 

0.127 

42.0 

300 

2.50 

0.167 

55.2 

400 

2.10 

0.187    . 

61.8 

450 

1.90 

0.191 

63.0 

500 

1.70 

0.189 

62.5 

600 

1.15 

0.153 

50.7 

700 

0.55 

0.086 

28.4 

775 

0.00 

0.000 

00.0 

2 

62.55 

0.081 

100 

6.70 

0.149 

25.9 

200 

6.10 

0.276 

48.0 

300 

5.40 

0.361 

62.7 

400 

4.70 

0.419 

72.8 

500 

3.85 

0.429 

74.5 

600 

3.00 

0.400 

69.5 

700 

1.80 

0.284 

49.2 

800 

0.70 

0.125 

21.6 

880 

0.00 

0.000 

00.0 

3 

62.58 

0.117 

100 

9.60 

0.214 

25.6 

200 

8.70 

0.388 

46.5 

300 

7.80 

0.522 

62.5 

400 

6.80 

0.606 

72.6 

500 

5.85 

0.652 

78.0 

600 

4.75 

0.635 

76.0 

700 

3.30 

0.515 

61.7 

800 

1.65 

0.294 

35.2 

930 

0.00 

0.000 

00.0 

4 

62.63 

0.150 

100 

12.30 

0.274 

25.6 

200 

11.25 

0.510 

47.6 

300 

10.05 

0.672 

62.8 

400 

8.85 

0.790 

73.8 

500 

7.80 

0.870 

81.3 

600 

6.45 

0.862 

80.6 

700 

4.75 

0.740 

69.1 

800 

2.60 

0.463 

42.3 

900 

0.85 

0  .  160 

15.0 

950 

0.00 

0.000 

00.0 

TEST  DATA 


271 


TABLE  8. — (Continued] 


Turns 
of 
needle 

Head, 
ft. 

Discharge, 
cU.  ft.  per 
sec. 

R.p.m. 

Brake 
load, 
Ib. 

B.h.p. 

Efficiency, 
per  cent. 

5 

62.68 

0.174 

100 

14.10 

0.314 

25.3 

200 

12.90 

0.575 

46.4 

300 

11.50 

0.768 

62.0 

400 

10.20 

0.908 

73.3 

500 

9.05 

1.010 

81.5 

600 

7.30 

0.975 

78.7 

700 

5.45 

0.850 

68.6 

800 

3.30 

0.515 

41.6 

900 

1.20 

0.241 

19.4 

970 

0.00 

0.000 

00.0 

6 

62.74 

0.200 

100 

15.25 

0.340 

23.8 

200 

14.00 

0.624 

43.7 

300 

12.65 

0.846 

59.5 

400 

11.25 

1.011 

71.0 

500 

9.95 

1.110 

78.0 

600 

8.20 

1.098 

77.0 

700 

6.00 

0.935 

65.6 

800 

3.60 

0.561 

39.4 

900 

1.20 

0.302 

21.2 

985 

0.00 

0.000 

00.0 

7.85 

62.81 

0.230 

100 

17.40 

0.388 

23.7 

200 

16.10 

0.717 

43.8 

300 

14.70 

0.982 

60.0 

400 

12.80 

1.140 

69.5 

500 

11.10 

1.235 

75.5 

600 

9.30 

1.242 

76.0 

700 

6.90 

1.080 

66.0 

800 

4.20 

0.655 

40.0 

900 

1.85 

0.370 

22.6 

985 

0.00 

0.000 

00.0 

272 


HYDRAULIC  TURBINES 


TABLE  9. — TEST  OF  A  12-iNCH  PELTON-DOBLE  TANGENTIAL  WATER 
WHEEL  UNDER  A  CONSTANT  PRESSURE  HEAD  OF  130.5  FT. 


Turns 
of 
needle 

Head, 
ft. 

Discharge, 
cu.  ft.  per 
sec. 

Brake 
R.p.m.      load, 
Ib. 

B.h.p. 

Efficiency, 
per  cent. 

1 

130.68 

0.062 

100 

6.8 

0.152 

16.4 

300 

6.2 

0.414 

44.8 

500 

5.2 

0.579 

62.8 

700 

4.0 

0.624 

67.6 

900 

2.2 

0.440 

47.7 

1100 

0.6 

0.147 

15.9 

1190 

0.0 

0.000 

00.0 

2 

130.74 

0.118 

100 

14.2 

0.316 

18.0 

300 

12.8 

0.855 

48.6 

500 

10.8 

1.201 

68.3 

700 

8.8 

1.372 

78.0 

900 

6.2 

1.242 

70.6 

1100 

3.2 

0.784 

44.5 

1300 

0.5 

0.144 

08.2 

1340 

0.0 

0.000 

00.0 

3 

130.83 

0.173 

100 

21.2 

0.472 

18.4 

300 

18.6 

1.242 

48.3 

500 

15.8 

1.760 

68.5 

700 

12.9 

2.010 

78.2 

800 

11.7 

2.080 

81.0 

900 

10.0 

2.002 

78.0 

1100 

5.4 

1.320 

51.4 

1300 

1.6 

0.398 

15.5 

1395 

0.0 

0.000 

00.0 

4 

130.95 

0.222 

100 

26.6 

0.592 

17.9 

300 

24.0 

1.602 

48.6 

500 

20.6 

2.295 

69.5 

700 

16.8 

2.620 

79.5 

800 

15.0 

2.670 

81.0 

900 

12.8 

2.660 

80.6 

1100 

7.6 

1.860 

56.5 

1300 

2.6 

0.752 

'     22.8 

1420 

0.0 

0.000 

00.0 

5 

131.06 

0.262 

100 

31.4 

0.700 

17.9 

300 

28.2 

1.884 

48.2 

< 

500 

24.2 

2.695 

69.0 

700 

19.5 

3.040 

78.0 

TEST  DATA 


273 


TABLE  9.— (Continued) 


Turns 
of 
needle 

Head, 
ft. 

Discharge, 
cu.  ft.  per 
sec. 

R.p.m. 

Brake 
load, 
Ib. 

Bh.p. 

Efficiency, 
per  cent. 

800 

17.6 

3.135 

80.3 

900 

15.2 

3.040 

78.0   ' 

1100 

8.8 

2.155 

55.2 

x,  - 

1300 

3.4 

0.984 

25.2 

1450 

0.0 

0.000 

00.0 

6 

131.19 

0.300 

100 

34.2 

0.762 

17.3 

300 

31.0 

2.070 

47.0 

500 

26.6 

2.960 

67.4 

700 

21.4 

3.335 

75.8 

800 

19.1 

3.400 

77.3 

v 

900 

16.5 

3.300 

75.0 

1100 

10.2 

2.497 

56.7 

1300 

4.2 

1.215 

27.6 

1460 

0.0 

0.000 

00.0 

7.85 

131  ,f4Q 

0.356 

100 

39.2 

0.874 

16.4 

300 

35.0 

2.340 

44.0 

500 

29.8 

3.315 

62.2 

700 

24.6 

3.830 

72.0 

800 

21.8 

3.880 

73.0 

. 

900 

18.7 

3.740 

70.3 

1100 

11.6 

2.838 

53.3 

1300 

4.8 

1.390 

26.2 

1460 

0.0 

0.000 

00.0 

274 


HYDRAULIC  TURBINES 


TABLE  10. — TEST    OF    A    12-iNCH    PELTON-DOBLE    TANGENTIAL    WATER 
WHEEL  UNDER  A  CONSTANT  PRESSURE  HEAD  OF  230   FT. 


Turns 
of 
nozzle 

Head, 

ft. 

Discharge, 
cu.  ft.  per 
sec. 

R.p.m. 

Brake 
load, 
Ib. 

B.h.p. 

Efficiency, 
per  cent. 

1 

230.2 

0.081 

100 

11.4 

0.254 

12.0 

300 

10.3 

0.687 

32.4 

500 

9.2 

1.025 

48.4- 

700 

7.8 

1.217 

57.5 

800 

7.0 

1.248 

59.0 

900 

6.2 

1.242 

58.7 

1100 

4.3 

1.054 

49.7 

1300 

2.0 

0.578 

27.3 

1440 

0.0 

0.000 

00.0 

2 

230.3 

0.163 

100 

24.6 

0.548 

12.8 

300 

22.3 

1.490 

34.9 

500 

19.7 

2.190 

51.2- 

700 

17.1 

2.665 

62.4 

900 

14.4 

2.880 

67.5 

1100 

11.4 

'2.795 

65.4 

1300 

8.0 

2.320 

54.3 

1500 

4.5 

1.505 

35.2 

1700 

1.0 

0.378 

12.5 

0.0 

0.000 

00.0 

3 

230.5 

0.231 

100 

31.3 

0.698 

11.5 

300 

32.7 

2.185 

36.2 

500 

28.3 

3.150 

52.1- 

700 

25.6 

3.990 

66.0 

900 

21.6 

4.330 

71.6 

1100 

17  A 

4.268 

70.6 

1300 

12.4 

3.588 

59.4 

1500 

7.0 

2.340 

38.7 

1700 

2.0 

0.756 

12.5 

1760 

0.0 

0.000 

00.0 

4 

230.6 

0.291 

100 

45.2 

1.005 

13.2 

300 

41.6 

2.780 

36.4 

500 

37.4 

4.160 

54.5- 

700 

33.0 

5.145 

67.4 

900 

28.2 

5.650 

74.0 

1100 

23.1 

5.665 

74.3 

1300 

17.6 

5.096 

66.7 

1500 

9.5 

3.180 

41.7 

1700 

3.9 

1.477 

19.3 

1790 

0.0 

0.000 

00.0 

TEST  DATA 


275 


TABLE  10. — (Continued) 


Turns 
of 
nozzle 

Head, 
ft. 

Discharge, 
cu.  ft.  per 
sec. 

R.p.m. 

Brake 
load, 
Ib. 

B.h.p. 

Efficiency, 
per  cent. 

5 

230.8 

0.343 

100 

54.2 

1.211 

13.5 

300 

49.6 

3.315 

36.8 

500 

44.4 

4.940 

54.9- 

700 

38.8 

6.050 

67.2 

900 

32.6 

6.530 

72.6 

1100 

26.6 

6.512 

72.4 

1300 

20.2 

5.850 

65.0 

1500 

12.8 

4.280 

47.6 

1700 

5.4 

2.040 

22.7 

1880 

0.0 

0.000 

00.0 

6 

231.1 

0.379 

100 

61.0 

1.360 

13.6 

300 

55.5 

3.710 

37.2 

500 

49.5 

5.510 

55.4- 

700 

43.0 

6.700 

67.3 

900 

36.4,, 

7.300 

73.4 

1000 

33.2 

7.390 

74.2 

1100 

29.7 

7.270 

73.0 

1300 

32.4 

6.480 

65.0 

1500 

14.4 

4.800 

48.2 

1700 

6.4 

2.420 

24.3 

1890 

0.0 

0.000 

00.0 

7.85 

231.2 

0.434 

100 

67.2 

1.499 

13.3 

300 

61.8 

4.130 

36.8 

500 

55.4 

6.160 

54.9- 

700 

47.8 

7.450 

66.3 

900 

40.0 

8.015 

71.4 

1000 

36.5 

8.125 

72.3 

1100 

32.9 

8.063 

71.8 

1300 

24.8 

7  .  180 

64.0 

1500 

16.3 

5.450 

48.5 

1700 

7.0 

2.550 

22.7 

1890 

0.0 

0.000 

00.0 

276 


HYDRAULIC  TURBINES 


TABLE  11. — TEST    OF    A    12-INCH    PELTON-DOBLE    TANGENTIAL    WATEH 
WHEEL  UNDER  A  CONSTANT  PRESSURE  HEAD  OF  305  FT. 


Turns 
of 
needle 

Head, 

ft. 

Discharge, 
cu.  ft.  per 
sec. 

R.p.m. 

Brake 
load, 
Ib. 

B.h.p. 

Efficiency, 
per  cent. 

1 

305.1 

0.1025 

0 

18.0 

0.00 

00.0 

400 

15.8 

1.41 

39.6 

800 

12.6 

2.24 

63.2 

1000 

10.8 

2.41 

67.7 

1200 

8.4 

2.24 

63.2 

14(30 

6.0 

1.87 

52.7 

1600 

3.2 

1.14 

32.2 

1800 

0.4 

0.16 

4.5 

1920 

0.0 

0.00 

0.0 

2 

305.2 

0.185 

0 

35.6 

0.00 

00.0 

400 

30.8 

2.74 

42.7 

800 

24.6 

4.38 

68.4 

1000 

21.0 

4.67 

72.7 

1200 

17.5 

4.67 

72.7 

1400 

13.4 

4.18 

65.1 

1600 

9.1 

3.24 

50.5 

1800 

4.4 

1.76 

27.4 

2020 

0.0 

0.00 

00.0 

3 

305.5 

0.278 

0 

52.8 

0.00 

00.0 

400 

45.2 

4.02 

41.5 

800 

36.6 

6.52 

67.4 

1000 

32.0 

7.12 

73.7 

1200 

26.8 

7.15 

74.0 

1400 

20.7 

6.45              66.8 

1600 

14.4 

5.13 

53.0 

1800 

7.8 

3.12 

32.2 

2080 

0.0 

0.00 

00.0 

4 

305.7 

0.341 

0 

62.8             0.00 

00.0 

400 

53.6             4.77 

40.3 

800 

43.0 

7.65 

64.7 

1000 

38.4 

*      8.55 

72.2 

1200 

33.0 

8.82 

74.5 

1400 

26.0 

8.10 

68.4 

1600 

18.6 

6.62 

55.9 

1800 

10.0 

4.00 

33.9 

2110 

00.0 

0.00 

00.0 

TEST  DATA 


277 


TABLE  11.— (Continued] 


Turns 
of 
needle 

Head, 
ft. 

Discharge, 
cu.  ft.  per 
sec. 

R.p.m. 

Brake 

load, 
Ib. 

B.h.p. 

Efficiency, 
per  cent. 

5 

306.0 

0.390 

0 

72.0 

0.00 

00.0 

400 

62.6 

5.57 

40.9 

800 

52.0 

9.27 

68.1 

1000 

45.8 

10.20 

75.0 

1200 

38.4 

10.25 

75.3 

1400 

30.2 

9.40 

69.0 

1600 

21.0 

7.48 

55.0 

1800 

11.0 

4.40 

32.4 

2150 

00.0 

0.00 

00.0 

TABLE    12.  —  FRICTION   AND   WINDAGE   OF   12-iNCH   PELTON-DOBLE    TAN 

GENTIAL 

WATER  WHEEL 

R.p.m. 

H.p. 

R.p.m. 

H.p. 

100 

0.0025 

800 

0.1545 

200 

0.0089 

900 

0.2190 

300 

0.0146 

1000 

0.2660 

400 

0.0305 

1100 

0.3270 

500 

0.0515 

1200 

0.3910 

600 

0.0746 

1300 

0.4980 

700 

0.1135 

1400 

0.5970 

1500 

0.7020 

TABLE  13. — TEST  OF  A  PELTON-DOBLE  TANGENTIAL  WATER  WHEEL  NEAR 

FRESNO,  CAL. 

Static  Head  =  1403.45  ft. 


Head, 
ft. 

Discharge,  cu. 
ft.  per  sec. 

H.p. 
input 

B.h.p. 

Efficiency, 
per  cent. 

1400.77                  17.50 

2790 

2075 

74.4 

1398.25                  22.10 

3510 

2767 

78.7 

1396.56                  27.00 

4280 

3450 

80.7 

1394.15 

31.90 

5050 

4120 

81.7 

1391.80 

36.70 

5800 

4765 

82.2 

1389.65 

42.00 

6630 

5480 

82.7 

1383.90 

54.00 

8475 

6825 

80.6 

is 


278 


HYDRAULIC  TURBINES 


TABLE  14. — TEST  OF  A  2|-iNCH  TWO-STAGE  WORTHINGTON  TURBINE  PUMP 
AT  CORNELL  UNIVERSITY 

By  R.  L.  Daugherty 

Diameter  of    Impellers  =  12   inches.     Test    made  at  a  constant  speed  of 

1700  r.p.m. 


Discharge,  cu.  ft. 
per  sec. 

Head, 
ft. 

B.h.p. 

Efficiency, 
per  cent. 

0.000 

248.5 

9.13 

00.0 

0.049 

248.6 

10.12 

13.7 

0.112 

254.6 

11.78 

27.6 

0  155 

257.5 

12.70 

35.8 

0.236 

264.1 

15.08 

47.1 

0.348 

248.8 

18.15 

54.3 

0.429 

225.0 

20.06 

55.0 

0.494 

192.2 

21.60 

50.0 

0.531 

157.3 

22.00 

43.2 

0.573 

73.1 

21.25 

22.4 

0.578 

47.1 

20.60 

15.0 

0.580 

26.3 

20.15 

8.6 

TABLE    15. — TEST  OF   A  6-lNCH   SINGLE-STAGE    DEL.AVAL   CENTRIFUGAL 
PUMP  AT  CORNELL  UNIVERSITY 

.     By  R.  L.  Daugherty 
Volute  Type.      Diameter  of  Impeller  =  9.11  inches.     Speed  1700  r.p.m. 


Discharge,  cu.  ft. 
per  sec. 

Head, 

ft. 

B.h.p. 

Efficiency, 
per  cent. 

0.000 

68.5 

4.3 

00.0 

0.068 

68.4 

4.5 

11.7 

0.188 

69.6 

5.2 

28.6 

0.320' 

69.3 

6.0 

42.0 

0.606 

69.2 

7.8 

61.0 

0.873 

65.8 

9.3 

70.2 

1.063 

62.7 

10.3 

73.5 

1.315 

55.7 

11.3 

73.7 

1.632 

47.3 

12.0 

73.2 

1.968 

35.7 

11.8 

67.7 

2.090 

28.1 

11.5 

58.0 

2.240 

22.3    . 

11.2 

50.7 

INDEX 


Absolute  velocity,  80 
Action  of  water,  7 
American  turbine,  9 
Annual  cost  of  power,  197 
Arrangement  of  runners,  11 
Axial  flow,  9 

B 

Barker's  mill,  41 
Bearings,  56 
Brake,  146 
Breast  wheel,  1 
Buckets,  31 

design  of,  208 

pitch  of,  205 


Capital  cost,  195 
Case,  58,  209 
Centrifugal  force,  97,  231 

pump,  230 

Chain  type  construction,  32 
Characteristic  curve,  173 
Classification  of  turbines,  7 
Clear  opening,  216 
Coefficient  of  nozzle,  114 
Conduit,  6 
Constants,  153 

determination  of,  160 

uses  of,  161 
Cost,  steam  power,  200 

turbines,  192 

water  power,  197 
Current  meter,  145 

wheel,  1 
Cylinder  gate,  52 


D 

Defects  of  theory/ 134,"  24  2 
Deflecting  nozzle,  36 
Diameter  of  runner,  52 

and  discharge,  154 

and  power,  155 

and  speed,  152 
Diffusion  vanes,  231 
Direction  of  flow,  8 
Discharge  curve,  15 

measurements  of,  143 
Doble  bucket,  30 
Draft  tube,  13,  63,  100 
Dynamometer,  146 


Eddy  loss,  70,  125 
Efficiency,  83 

as  function  of  speed  and  gate, 
164 

of  impulse  wheel,  39,  115 

maximum,  177 

on  part  load,  179 

of  reaction  turbine,  72 

relative,  182 
Energy,  80 


Falling  stream,  effect  on  flow,  15 

Fitz  water  wheel,  2 

Flow  and  head,  18 

Flow,  direction  of,  8 

Flume,  14 

Force  exerted,  84,  108 

Forebay,  6 

Fourneyron  turbine,  8,  41,  93 

Francis  turbine,  3,  41 


279 


280 


INDEX 


Friction,  147 

variation  with  speed,  151 
Full  gate,  183 
Full  load,  179 


G 


Gage  heights,  fluctuations  in,  16 

relation  of,  to  flow,  15 
Gates,  cylinder,  52 

design  of,  228 

register,  53 

wicket,  53 

Girard  turbine,  30,  92 
Governing — see  speed  regulation. 
Governors,  74 
Gradient,  15 
Guides — see  gates. 

H 

Hand  of  turbine,  9 
Head,  -80 

delivered  to  runner,  91 

and  discharge,  152 

and  efficiency,  152 

measurement  of,  143 

net  for  pump,  233 

net  for  turbine,  81 

and  power,  152 

and  speed,  152 

variation  of,  17 

-water,  6 

Holyoke  testing  flume,  140 
Howd  turbine,  41 
Hydraucone,  68 
Hydraulic  gradient,  21 
Hydrograph,  16 


1 


Impending  delivery,  237 
Impulse  circle,  204 
Impulse  turbine,  7 

advantages  of,  180,  190 

problem  of,  92 
Inward  flow,  3,  8 

advantages  of,  46 


Jet,  use  of  two  or  more,  34 

size  of,  to  size  of  wheel,  33 


Link,  55 
Load  "curve,  19 

factor,  19 

factor  and  cost,  198 
Losses,  centrifugal  pump,  234 

impulse  turbine,  93,  113 

reaction  turbine,  94,  125 

M 

Manoinetric  efficiency,  233 
McCormick  runner,  42 
Mixed  flow,  9 

N 

Needle  nozzle,  35 
Nozzle  coefficients,  114 
design,  204 

O 

Operating  expenses,  197 
Outward  flow  turbine,  8 
Overgate,  183 
Overhung  wheel,  12 
Overload,  180 
Overshot  wheel,  2 


Part  load,  180 

Pelton  wheel,  30 

Penstock,  6 

Pitot  tube,  144 

Pivoted  guides,  53 

Pondage,  19 

Power,  83 

delivered  to  runner,  91 
effect  of  head  on,  18 
effect  of  pondage  on,  19 
effect  of  storage  on,  20 
measurement  of,  146 
through  pipe  line,  21 
of  stream,  19 


INDEX 


281 


R 


Race,  6 
Radial  flow,  8 
Rainfall  and  run-off,  16 

use  of  records  of,  17 
Rating  curve,  15 
Reaction  turbine,  7 

advantages  of,  179,  185,  189 

development  of,  41 

problem  of,  93 
Register  gate,  53 
Regulation,  52 
Relative  velocity,  80 
Relief  valve,  24,  37,  56 
Retardation  runs,  148,  249 
Rheostats,  146 
Runners,  types  of,  43 

construction  of,  51 

design  of,  211 

diameter  of,  52 


Sale  of  power,  198 
Scotch  turbine,  41 
Selection  of  turbine,  169,  177 
Shaft,  choice  of,  10 

position  of,  9 
Shifting  rim,  55 
Shock  loss,  125,  236 
Shut-off  head,  237 
Specific  speed,  155,  167,  245 
Speed,  effect  of,  94 

ring,  61,  227 
Speed  regulation,  35,  52 

effect  of  pipe  on,  23 
Splitter,  32 
Stand  pipe,  24 
Steam  power  cost,  200 
Step  bearing,  56 
Storage,  20 


Stream  flow,  16 
gaging,  15 

Suction  tube — see  draft  tube. 
Surge  chamber,  24 
Suspension  bearing,  57 
Swain  turbine,  42 


Tail  race,  6 

Tangential  water  wheel,  105 

Testing,  140 

Thrust  of  runnet,  58 

Torque,  89 

Turbines,  advantages  of,  4 

classification  of,  7 

definition  of,  2 

types  of,  7 
Turbine  pump,  230 

U 

Undershot  water  wheel,  1 
V 

Value  of  water  power,  202 
Vanes,  number  of,  214 

layout  of,  219 
Velocity  in  case,  227 

of  pump  discharge,  232 
Volute  pump,  230 
Vortex,  97 

W 

Water  hammer,  24 
Water,  measurement  of,  15 
Water  power,  15 
Water  power  plant,  5 
Water  wheels,  1,  3,  4 
Wear  of  turbines,  70,  102 
Wicket  gate,  53 


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